The exponent tells you how many times to use the base as a factor.
Step-by-step solution
The base is 3 and the exponent is 4, so 3 is used as a factor 4 times.
Write it out: 3^4 = 3 \times 3 \times 3 \times 3.
Multiply step by step: 3 \times 3 = 9, then 9 \times 3 = 27, then 27 \times 3 = 81.
Answer: 81.
Check:3^2 = 9 and 3^4 = (3^2)^2 = 9^2 = 81. ✓
Tip: An exponent is repeated multiplication, not multiplication of the base by the exponent. 3^4 \ne 3 \times 4, and 3^4 \ne 4^3 — the order of base and exponent matters.
Question 2 of 36
No calculator
Which expression is equivalent to x^5 \cdot x^3?
Hint
Write out each power as repeated factors and count how many x's you have in total.
Step-by-step solution
x^5 = x \cdot x \cdot x \cdot x \cdot x (5 factors) and x^3 = x \cdot x \cdot x (3 factors).
Multiplying puts all the factors together: 5 + 3 = 8 factors of x.
So x^5 \cdot x^3 = x^{5+3} = x^8.
This is the product rule: same base, multiply → add the exponents.
Answer: x^8.
Check: try x = 2: 2^5 \cdot 2^3 = 32 \cdot 8 = 256, and 2^8 = 256. ✓
Tip: Add exponents when you multiply powers of the same base. Multiply exponents only for a power of a power, like (x^5)^3 = x^{15}.
Question 3 of 36
No calculatorFill in
What is the value of 4^{-2}? Write your answer as a fraction or a decimal.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
A negative exponent does not make the answer negative. It means "one over" the positive power.
Tip: Common wrong answers are -16 (making the result negative) and -8 (multiplying 4 \times -2). A negative exponent means reciprocal: the answer is a small positive number.
Question 4 of 36
No calculator
Without a calculator, find -2^4. Notice that there are no parentheses around the -2.
Hint
Check what the exponent is attached to. Is the negative sign inside parentheses with the 2, or not?
Step-by-step solution
Order of operations: exponents come before the negative sign (which acts like multiplying by -1).
So -2^4 means -(2^4).
2^4 = 2 \times 2 \times 2 \times 2 = 16.
Apply the negative: -(16) = -16.
Answer: -16.
Check: compare with (-2)^4 = (-2)(-2)(-2)(-2) = 16. The parentheses are what make the -2 the base. Without them, only the 2 is raised to the power. ✓
Tip:-a^n and (-a)^n are different. If there are no parentheses, the exponent touches only the number right next to it. On the TI-30XS, typing (−) 2 ^ 4 also gives -16.
Question 5 of 36
No calculatorFill in
Write the value of \dfrac{10^{9}}{10^{6}} as a whole number.
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Both powers have the same base, 10. What do you do with the exponents when you divide?
Step-by-step solution
Quotient rule: \dfrac{a^m}{a^n} = a^{m-n} when the bases are the same.
Here: \dfrac{10^9}{10^6} = 10^{9-6} = 10^3.
10^3 = 10 \times 10 \times 10 = 1{,}000.
Answer: 1,000.
Check:10^9 is one billion (1{,}000{,}000{,}000) and 10^6 is one million (1{,}000{,}000). One billion divided by one million is 1{,}000. ✓
Tip: For powers of ten, the exponent counts zeros. Dividing removes zeros: 9 zeros minus 6 zeros leaves 3 zeros. A common wrong answer is 10^{1.5} (dividing the exponents 9 \div 6) — you subtract, not divide.
Question 6 of 36
No calculator
What is the value of 5^{0} + 5^{-1}?
Hint
Find each term on its own. Any nonzero number to the power 0 is the same value. A negative exponent means "one over".
Step-by-step solution
Zero exponent: 5^0 = 1 (any nonzero base to the power 0 equals 1).
Tip: The pattern "divide by the base each time the exponent drops by 1" explains both rules at once: a^0 = 1 and a^{-1} = \tfrac1a. Neither rule ever produces a negative number.
Question 7 of 36
No calculator
A strand of human hair is about 0.00007 meter wide. How is this width written in scientific notation?
Hint
Move the decimal point until exactly one nonzero digit is in front of it. Did you move it left or right?
Step-by-step solution
The first nonzero digit is 7. Scientific notation needs a number from 1 up to (not including) 10, so the front number is 7.
Count the places the decimal point moves from 0.00007 to 7.: it passes the digits 0, 0, 0, 0, 7 — that is 5 places to the right.
Moving right for a small number gives a negative exponent: 10^{-5}.
Tip: Numbers less than 1 always have a negative exponent in scientific notation. Count decimal places moved, not zeros: 0.00007 shows four zeros after the point, but the point moves five places.
Question 8 of 36
No calculatorFill in
Find the value of n that makes the equation true.
x^{7} \cdot x^{n} = x^{12}
n =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
When you multiply powers with the same base, what happens to the exponents? Write an equation using only the exponents.
Step-by-step solution
Product rule: x^7 \cdot x^n = x^{7+n}.
The equation says x^{7+n} = x^{12}. Same base, so the exponents must be equal: 7 + n = 12.
Tip: Turning a power equation into an exponent equation is the key move in "find the missing exponent" questions. A common wrong answer is \tfrac{12}{7}, which uses the multiplying rule instead of the adding rule.
Question 9 of 36
No calculator
Which expression is equivalent to (2x^3)^4?
Hint
The outside exponent applies to every factor inside the parentheses — including the number 2.
Tip: Do not just count zeros — count decimal places moved. 93{,}000{,}000 has 6 zeros, but the point moves 7 places because the digit 3 also gets passed.
Question 11 of 36
Fill in
A certain bacterium is 2 \times 10^{-6} meter long. A certain virus is 4 \times 10^{-8} meter long.
How many times longer is the bacterium than the virus?
times
Type your answer in the box. Fractions like 3/4 are fine.
Hint
"How many times longer" means divide the larger length by the smaller one. Divide the front numbers, then subtract the exponents.
Step-by-step solution
Set up the division: \dfrac{2 \times 10^{-6}}{4 \times 10^{-8}}.
Divide the front numbers: \dfrac{2}{4} = 0.5.
Divide the powers of ten by subtracting exponents: \dfrac{10^{-6}}{10^{-8}} = 10^{-6 - (-8)} = 10^{2}.
Tip: Watch the double negative: -6 - (-8) = -6 + 8 = 2. Writing -6 - 8 = -14 is the most common error here. Also, "times longer" always means divide — subtracting gives a difference, not a ratio.
Question 12 of 36
No calculatorDrop-down
Select the expression that makes the statement true. (Assume x \ne 0.)
x^{-3} is equivalent to .
Choose an answer from each menu.
Hint
Use x^{-3} \cdot x^{3} = x^0 = 1. What do you multiply x^3 by to get 1?
Step-by-step solution
Negative exponent rule: a^{-n} = \dfrac{1}{a^n} (for a \ne 0).
Apply it with a = x and n = 3: x^{-3} = \dfrac{1}{x^3}.
Why it works: x^{-3} \cdot x^3 = x^{-3+3} = x^0 = 1, so x^{-3} is the reciprocal of x^3.
Answer: \dfrac{1}{x^3}.
Check: let x = 2: 2^{-3} = \tfrac{1}{2^3} = \tfrac18 = 0.125. The other choices give -8, -6 and about 1.26 — none equal 0.125. ✓
Tip: A negative exponent moves the factor across the fraction bar and makes the exponent positive. It never makes the value negative by itself.
Question 13 of 36
No calculator
What is the value of 27^{\frac{2}{3}}?
Hint
In a^{m/n}, the bottom number n is a root and the top number m is a power. Take the root first to keep numbers small.
Step-by-step solution
Rule: a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m. Here m = 2 (power) and n = 3 (cube root).
Tip: Exponent rules work for multiplying and dividing powers with the same base. For adding powers, you must calculate each one first.
Question 15 of 36
No calculator
A streaming service has about 3 \times 10^{4} subscribers in one city. Each subscriber streams about 2.5 \times 10^{3} minutes of video per month.
About how many minutes of video do these subscribers stream in a month, in total?
Hint
A total for many people who each do the same amount means multiply. Multiply the front numbers, and combine the powers of ten with the product rule.
Step-by-step solution
Total = (number of subscribers) \times (minutes each): (3 \times 10^4)(2.5 \times 10^3).
Multiply the front numbers: 3 \times 2.5 = 7.5.
Multiply the powers of ten by adding exponents: 10^4 \times 10^3 = 10^{4+3} = 10^7.
Combine: 7.5 \times 10^7. Since 7.5 is between 1 and 10, this is already in scientific notation.
Answer: 7.5 \times 10^7 minutes.
Check: in standard form, 30{,}000 \times 2{,}500 = 75{,}000{,}000, and 7.5 \times 10^7 = 75{,}000{,}000. ✓
Tip: Multiply the front numbers, add the exponents. Never add the front numbers (3 + 2.5) and never multiply the exponents (4 \times 3).
Question 16 of 36
No calculatorDrop-down
Simplify \left(x^{2}y^{5}\right)^{3}. Select the correct exponents.
The exponent of x is , and the exponent of y is .
Choose an answer from each menu.
Hint
The outside exponent 3 applies to each factor inside the parentheses. For a power of a power, do you add or multiply?
Step-by-step solution
Power of a product: \left(x^2y^5\right)^3 = \left(x^2\right)^3 \cdot \left(y^5\right)^3.
Power of a power — multiply exponents:
\left(x^2\right)^3 = x^{2 \times 3} = x^6
\left(y^5\right)^3 = y^{5 \times 3} = y^{15}
Result: x^6y^{15}.
Answers: the exponent of x is 6; the exponent of y is 15.
Check: write it out: \left(x^2y^5\right)^3 = (x^2y^5)(x^2y^5)(x^2y^5). Now use the product rule: x^{2+2+2} = x^6 and y^{5+5+5} = y^{15}. ✓
Tip: Adding exponents is for multiplying separate powers (x^2 \cdot x^3). A power of a power multiplies the exponents. The exponent itself is never raised to a power.
Question 17 of 36
No calculator
Dana simplified \dfrac{6x^{8}}{2x^{2}} (with x \ne 0) as shown.
Step 1:\dfrac{6}{2} = 3
Step 2:\dfrac{x^{8}}{x^{2}} = x^{4}
Step 3: So the expression equals 3x^{4}.
Which statement is true?
Hint
Check each step on its own. Which rule should be used to divide powers with the same base?
Step-by-step solution
Step 1 divides the coefficients: 6 \div 2 = 3. This is correct.
Step 2 divides powers with the same base. The quotient rule says subtract the exponents: \dfrac{x^8}{x^2} = x^{8-2} = x^6. Dana divided the exponents (8 \div 2 = 4), so Step 2 is wrong.
Step 3 should therefore be 3x^6.
Answer: Step 2 is wrong; it should be x^6, so the result is 3x^6.
Tip: Coefficients and exponents follow different rules. Coefficients are divided normally; exponents of the same base are subtracted. Plugging in a small number like x = 2 is a fast way to catch an error.
Question 18 of 36
No calculator
In computer memory, 1 kilobyte is 2^{10} bytes, and 1 megabyte is 2^{10} kilobytes.
How many bytes are in 1 megabyte?
Hint
To change megabytes to bytes, you multiply by the number of kilobytes in a megabyte and then by the number of bytes in a kilobyte.
Step-by-step solution
Bytes in 1 megabyte = (kilobytes per megabyte) \times (bytes per kilobyte).
Answer: 2^{20} bytes (which is 1{,}048{,}576 bytes).
Check:2^{10} = 1{,}024. Then 1{,}024 \times 1{,}024 = 1{,}048{,}576, and 2^{20} = 1{,}048{,}576. ✓
Tip: Keep the base the same and add the exponents. Do not multiply the bases (2 \times 2 = 4) — the base stays 2. And do not add the powers as if they were plain numbers: 2^{10} + 2^{10} = 2^{11}, which is a sum, not a product.
Question 19 of 36
No calculatorFill in
What is the value of 16^{\frac{3}{4}}?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The denominator of the exponent tells you which root to take. Take that root of 16 first, then apply the numerator as a power.
Step-by-step solution
Rule: a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m. Here n = 4 (fourth root) and m = 3 (cube).
Fourth root of 16: \sqrt[4]{16} = 2, because 2 \times 2 \times 2 \times 2 = 16.
Cube the result: 2^3 = 8.
Answer: 8.
Check: write 16 as a power of 2: 16 = 2^4, so 16^{3/4} = \left(2^4\right)^{3/4} = 2^{4 \cdot \frac34} = 2^3 = 8. ✓
Tip: Root first, power second keeps the numbers small (16^3 = 4{,}096 is much harder to take a fourth root of). Wrong answers to watch for: 12 (from 16 \times \tfrac34) and 2 (root taken but not cubed). On the TI-30XS you can type 16 ^ ( 3 ÷ 4 ).
Question 20 of 36
No calculatorSelect all
Which of these expressions have a value of exactly \dfrac{1}{9}? Choose every one that does.
Select all that apply.
Hint
Rewrite each choice as a single power of 3 (or work out its value). A negative exponent means reciprocal.
Step-by-step solution
Target: \dfrac19 = \dfrac{1}{3^2} = 3^{-2}. Test each choice:
3^{-2} = \dfrac{1}{3^2} = \dfrac19 ✓
-3^{2} = -(3^2) = -9 ✗ (negative, not a reciprocal)
9^{-1} = \dfrac{1}{9} ✓
\dfrac{3^4}{3^6} = 3^{4-6} = 3^{-2} = \dfrac19 ✓
\left(\tfrac13\right)^{-2} = \left(\tfrac31\right)^{2} = 9 ✗ (a negative exponent flips the fraction)
Tip: A negative exponent on a fraction flips the fraction: \left(\tfrac{a}{b}\right)^{-n} = \left(\tfrac{b}{a}\right)^n. And -3^2 is not \tfrac19 — the minus sign is outside the power.
Question 21 of 36
No calculatorDrag and drop
A science class measured the widths of four particles, in millimeters. Drag each width into the table to order them from least to greatest.
Least
2nd
3rd
Greatest
Tap a tile, then tap a box to place it.
Hint
Compare the exponents first: with negative exponents, the number closer to zero (like -2) is the larger power of ten. Only compare front numbers when the exponents match.
Step-by-step solution
Compare the powers of ten first. For negative exponents, 10^{-2} > 10^{-3} > 10^{-4} (hundredths are bigger than thousandths).
Smallest power: 5 \times 10^{-4} is the only one with 10^{-4}, so it is the least (0.0005).
Next: 4.2 \times 10^{-3} = 0.0042.
Two numbers share 10^{-2}; compare their front numbers: 1.1 < 3.9, so 1.1 \times 10^{-2} = 0.011 comes before 3.9 \times 10^{-2} = 0.039.
Tip: Do not order by the front numbers alone (1.1, 3.9, 4.2, 5) — that ignores the exponents, which matter much more. With negative exponents, the more negative exponent is the smaller number.
Question 22 of 36
A community center uses a phone tree to share news. In round 1, the coordinator calls 3 volunteers. In each later round, every person called in the previous round calls 3 new people.
How many people are called in round 5 alone?
Hint
Make a short table: round 1, round 2, round 3 … How does each round's number compare with the one before it?
Step-by-step solution
Round 1: 3 people = 3^1.
Round 2: each of those 3 calls 3 people: 3 \times 3 = 9 = 3^2.
Round 3: 9 \times 3 = 27 = 3^3.
The pattern: round r has 3^r calls. So round 5 has 3^5.
Tip: "Each one does it again" situations grow by repeated multiplication — that is an exponent. Read carefully: "in round 5 alone" is 243, but "in all five rounds" would be 3 + 9 + 27 + 81 + 243 = 363.
Question 23 of 36
Fill in
A data center has 6.4 \times 10^{12} bytes of free storage. A high-definition movie file uses about 3.2 \times 10^{9} bytes.
How many movie files of this size fit in the free storage?
files
Type your answer in the box. Fractions like 3/4 are fine.
Hint
"How many fit" means divide the total by the size of one. Divide the front numbers and subtract the exponents.
Step-by-step solution
Set up the division: \dfrac{6.4 \times 10^{12}}{3.2 \times 10^{9}}.
Front numbers: 6.4 \div 3.2 = 2.
Powers of ten: 10^{12} \div 10^{9} = 10^{12-9} = 10^3.
Tip: On the TI-30XS, wrap each number in parentheses: (6.4×10^12▶)÷(3.2×10^9▶) (the right arrow leaves the exponent). Without the parentheses the calculator may divide by 3.2 and then multiply by 10^9. Common wrong answers: 2 \times 10^{21} (adding exponents) and 3.2 (subtracting front numbers).
Question 24 of 36
No calculator
Which expression is equivalent to \dfrac{\left(3x^{-2}y^{4}\right)^2}{x^{3}y^{5}}? (Assume x \ne 0 and y \ne 0.)
Hint
Work in order: apply the outside exponent to every factor, then divide powers with the same base by subtracting exponents.
Step-by-step solution
Power of a product — square every factor in the numerator:
\left(3x^{-2}y^{4}\right)^2 = 3^2 \cdot x^{-2 \cdot 2} \cdot y^{4 \cdot 2} = 9x^{-4}y^{8}.
Quotient rule — subtract exponents for each base:
x: x^{-4} \div x^{3} = x^{-4-3} = x^{-7}
y: y^{8} \div y^{5} = y^{8-5} = y^{3}
Result so far: 9x^{-7}y^{3}.
Negative exponent — move x^{-7} to the denominator: \dfrac{9y^{3}}{x^{7}}.
Answer: \dfrac{9y^3}{x^7}.
Check: let x = 1, y = 2: original = \dfrac{(3 \cdot 1 \cdot 16)^2}{1 \cdot 32} = \dfrac{2304}{32} = 72; answer = \dfrac{9 \cdot 8}{1} = 72. ✓
Tip: Do one rule at a time and write each step. Most errors come from multiplying the coefficient by the exponent (3 \times 2) or losing the sign of a negative exponent.
Question 25 of 36
Fill in
Light travels at about 3.0 \times 10^5 kilometers per second. The Sun is about 1.5 \times 10^8 kilometers from Earth.
About how many minutes does it take sunlight to reach Earth? Round to the nearest tenth of a minute.
minutes
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Time = distance \div speed. That gives seconds; then convert seconds to minutes.
Check:8.3 \text{ min} \times 60 = 498 s, and 498 \times 3.0 \times 10^5 \approx 1.49 \times 10^8 km — very close to 1.5 \times 10^8. ✓
Tip: Track units in a multi-step problem. Stopping at 500 answers in seconds; the question asks for minutes. On the calculator, put each number in parentheses: (1.5×10^8▶)÷(3×10^5▶)enter (the right arrow leaves the exponent).
Question 26 of 36
No calculator
Simplify completely, using only positive exponents in your result (assume a \ne 0 and b \ne 0):
Tip: Write the subtraction with parentheses when an exponent is negative: -3 - (-4). Dropping the parentheses (-3 - 4 = -7) is the most common error on this kind of item.
Question 27 of 36
Fill in
Healthy blood contains about 5 \times 10^{6} red blood cells per microliter. There are 1{,}000 microliters in 1milliliter.
About how many red blood cells are in a 4.5-milliliter blood sample? Write your answer in scientific notation.
Front number: \quad Exponent of 10:
Type your answer in the box. Fractions like 3/4 are fine.
Hint
First change 4.5 mL into microliters. Then multiply by the number of cells in each microliter. Make sure your front number is between 1 and 10.
Step-by-step solution
Convert the sample to microliters: 4.5 \text{ mL} \times 1{,}000 = 4{,}500 \text{ µL} = 4.5 \times 10^3 µL.
Tip: Two common slips: forgetting the mL-to-µL conversion (giving 2.25 \times 10^7), and leaving the answer as 22.5 \times 10^9, which is not scientific notation because 22.5 > 10.
Question 28 of 36
No calculator
What value of n makes this equation true? (Assume x \ne 0.)
\dfrac{x^{n} \cdot x^{2}}{x^{9}} = x^{-3}
Hint
Simplify the left side to a single power of x that still contains n. Then set its exponent equal to -3.
Tip: Collapse the left side into one exponent expression first (n - 7), then solve a simple one-step equation. Keep track of the sign of every exponent: the 2 in the numerator adds, the 9 in the denominator subtracts.
Question 29 of 36
No calculator
Which expression is equivalent to \left(\dfrac{3x^{2}}{y^{3}}\right)^{4}? (Assume y \ne 0.)
Hint
The outside exponent applies to every factor on the top and the bottom of the fraction, including the number 3.
Step-by-step solution
Power of a quotient: \left(\dfrac{3x^2}{y^3}\right)^4 = \dfrac{\left(3x^2\right)^4}{\left(y^3\right)^4}.
Check: let x = 1, y = 1: original = 3^4 = 81 and answer = 81 ✓. Let x = 2, y = 1: original = (12)^4 = 20{,}736 and 81 \cdot 2^8 = 81 \cdot 256 = 20{,}736 ✓.
Tip:3^4 = 3 \times 3 \times 3 \times 3 = 81, not 3 \times 4 = 12. Every factor inside the parentheses — number, top variable, bottom variable — gets the power.
Question 30 of 36
No calculator
Luis divided \left(8 \times 10^{5}\right) \div \left(4 \times 10^{-2}\right) as shown.
Step 1:8 \div 4 = 2
Step 2:10^{5} \div 10^{-2} = 10^{3}
Step 3: The answer is 2 \times 10^{3}.
Which statement is true?
Hint
Use the quotient rule in Step 2 and write the subtraction carefully with parentheses.
Step-by-step solution
Step 1:8 \div 4 = 2. Correct.
Step 2: quotient rule: \dfrac{10^5}{10^{-2}} = 10^{5-(-2)} = 10^{5+2} = 10^7. Luis wrote 10^3 — he computed 5 - 2 and lost the double negative. Step 2 is wrong.
Correct answer:2 \times 10^7.
Answer: Step 2 is wrong; the correct quotient is 2 \times 10^7.
Check: in standard form: 800{,}000 \div 0.04 = 20{,}000{,}000 = 2 \times 10^7 ✓. (Dividing by a small number like 0.04 must give a larger result, so 2{,}000 was too small.)
Tip: Dividing by a negative power of ten makes the result bigger: \dfrac{1}{10^{-2}} = 10^2. Write 5 - (-2) with parentheses so you see the two minus signs.
Question 31 of 36
No calculatorDrop-down
Kim simplified \left(-2x^{3}\right)^{2} as shown.
Step 1:(-2)^{2} = -4
Step 2:\left(x^{3}\right)^{2} = x^{6}
Step 3: So \left(-2x^{3}\right)^{2} = -4x^{6}.
The first error is in . The correct simplified expression is .
Choose an answer from each menu.
Hint
What is (-2) \times (-2)? Then check the exponent rule used in Step 2.
Step-by-step solution
Step 1:(-2)^2 = (-2)(-2) = +4, because a negative times a negative is positive. Kim wrote -4, so Step 1 is the first error.
Step 2: power of a power: \left(x^3\right)^2 = x^{3 \cdot 2} = x^6. This step is correct.
Correct result:4x^6.
Answers: the first error is in Step 1; the correct expression is 4x^6.
Check: let x = 1: \left(-2 \cdot 1\right)^2 = (-2)^2 = 4, and 4 \cdot 1^6 = 4 ✓. Kim's -4x^6 gives -4 ✗. Any even power of a real number is never negative.
Tip: Inside parentheses, the negative sign is part of the base, so it is squared too. An even power always gives a non-negative result.
Question 32 of 36
No calculatorSelect all
More than one answer is correct. Mark each expression whose value matches 8^{\frac{2}{3}}.
Select all that apply.
Hint
Evaluate 8^{2/3} first: the denominator 3 is a cube root, the numerator 2 is a square. Then find the value of each choice.
Put it together: \dfrac{8 \cdot \frac12}{\frac14} = \dfrac{4}{\frac14} = 4 \times 4 = 16.
Answer: 16.
Check (all powers of 2):4^{3/2} = 2^3, 8^{-1/3} = 2^{-1}, so the expression is \dfrac{2^{3} \cdot 2^{-1}}{2^{-2}} = 2^{3 - 1 - (-2)} = 2^4 = 16. ✓
Tip: Dividing by a negative power multiplies by the positive power: \dfrac{1}{2^{-2}} = 2^2. Rewriting every base as a power of the same number (here 2) turns the problem into simple exponent arithmetic.
Question 34 of 36
A large cargo ship carries about 2.4 \times 10^{4} containers. Each loaded container has a mass of about 2.5 \times 10^{4} kilograms.
What is the approximate total mass of the containers in metric tons? (1 metric ton = 10^{3} kilograms.)
Hint
Find the total mass in kilograms first. Then decide: to change kilograms into the bigger unit, metric tons, do you multiply or divide by 10^3?
Step-by-step solution
Total mass in kilograms: (2.4 \times 10^4)(2.5 \times 10^4).
Front numbers: 2.4 \times 2.5 = 6.
Powers: 10^{4+4} = 10^8.
Total: 6 \times 10^8 kg.
Convert to metric tons. A ton is bigger than a kilogram, so there are fewer tons: divide by 10^3.
\dfrac{6 \times 10^8}{10^3} = 6 \times 10^{8-3} = 6 \times 10^5 metric tons.
Answer: about 6 \times 10^5 metric tons (600,000 tons).
Check:24{,}000 \times 25{,}000 = 600{,}000{,}000 kg, and 600{,}000{,}000 \div 1{,}000 = 600{,}000 t = 6 \times 10^5 t. ✓
Tip: When you change to a larger unit, the number gets smaller (divide). Before choosing an answer, ask whether its size makes sense: 6 \times 10^{11} tons would be a million times the answer — far more than any ship could carry.
Question 35 of 36
No calculator
A negative exponent sits outside the parentheses in the numerator below. Rewrite the whole quotient in simplest form with positive exponents only (m and n are nonzero).
Check: let m = 1, n = 1: original = \dfrac{4^{-2}}{8} = \dfrac{1/16}{8} = \dfrac{1}{128} ✓. Let m = 1, n = 2: numerator = (4 \cdot 4)^{-2} = \tfrac{1}{256}; denominator = 8 \cdot \tfrac{1}{32} = \tfrac14; original = \tfrac{4}{256} = \tfrac{1}{64}; answer = \tfrac{2}{128} = \tfrac{1}{64} ✓.
Tip: A negative outside exponent flips every factor's exponent sign and turns the coefficient into a reciprocal. Do the coefficient separately, then each variable, one rule at a time.
Tip: When the product of the front numbers is 10 or more, one power of ten moves into the exponent. Forgetting that shift gives a = 7, which is off by a factor of 10.
