SHSAT: Exponent rules and scientific notation
Use the product, quotient and power rules, handle zero and negative exponents, and compute with numbers in scientific notation.
How do you simplify expressions with exponents, including negative exponents and powers of 10, without a calculator?
Short answer
With the same base, add exponents when multiplying, subtract when dividing, and multiply for a power of a power. A negative exponent means a reciprocal.
Know first: Multiplying and dividing whole numbers, Place value with powers of 10
Step 1 of 5
Same base: add, subtract or multiply the exponents
An exponent counts repeated factors: 2^4 = 2 × 2 × 2 × 2. Every exponent rule comes from counting those factors. Multiplying 2^3 by 2^4 strings together 3 + 4 = 7 twos.
A negative exponent means a reciprocal, not a negative number: 2^(−3) = 1/2^3 = 1/8. Any nonzero number to the power 0 is 1.
| Rule | Example |
|---|---|
| a^m × a^n = a^(m + n) | 3^2 × 3^5 = 3^7 |
| a^m ÷ a^n = a^(m − n) | 7^6 ÷ 7^4 = 7^2 = 49 |
| (a^m)^n = a^(m × n) | (2^3)^2 = 2^6 = 64 |
| a^0 = 1 | 9^0 = 1 |
| a^(−n) = 1/a^n | 4^(−2) = 1/16 |
| a^2 + a^2 = 2a^2 | not a^4: 3^2 + 3^2 = 18, but 3^4 = 81 |
Rules for multiplying and dividing need the same base. Adding powers has no shortcut.
Step 2 of 5
Apply one rule at a time
Worked example: Power of a power, then a quotient
What is the value of (10^2)^3 ÷ 10^4?
- A3/2
- B10
- C100
- D10^10
- 1
Power of a power
(10^2)^3 = 10^(2 × 3) = 10^6.
- 2
Quotient with the same base
10^6 ÷ 10^4 = 10^(6 − 4) = 10^2.
- 3
Evaluate
10^2 = 100. Check: (10^2)^3 = 1,000,000, and 1,000,000 ÷ 10,000 = 100.
- 4
See why the other choices are there
3/2 divides the exponents, 6 by 4. 10 adds 2 + 3 instead of multiplying, giving 10^5 ÷ 10^4. 10^10 adds the exponents in the quotient instead of subtracting.
Answer
100
Find the wrong step
Evaluate 5^0 + 4^(−2).
- 1
5^0 = 1.
- 2
4^(−2) = −16.
What went wrong
A negative exponent means a reciprocal. It doesn't make the number negative.
- 3
1 + (−16) = −15.
The fix
4^(−2) = 1/4^2 = 1/16, so the sum is 1 + 1/16 = 17/16.
Step 3 of 5
Rules, notation and look-alikes
Check yourself · Question 1
What is the value of (5^2)^3 ÷ 5^5?
Answer: C
(5^2)^3 = 5^6. Then 5^6 ÷ 5^5 = 5^1 = 5.
Check yourself · Question 2
Which is equal to (9 × 10^7) ÷ (3 × 10^(−2))?
Answer: B
Divide the front numbers: 9 ÷ 3 = 3. Subtract the exponents: 7 − (−2) = 9. The result is 3 × 10^9.
Check yourself · Question 3
Which expression is equal to 16?
Answer: D
2^6 ÷ 2^2 = 2^4 = 16. The others are −16, 8 and 8.
Trap answer: 2^2 + 2^2
Adding exponents when adding powers
- Why it looks right
- It looks like 2^2 × 2^2, which really is 2^4 = 16.
- Why it's wrong
- The exponent rules apply to multiplying, not adding. 2^2 + 2^2 is two fours, which is 8.
- The right answer
- 2^6 ÷ 2^2, which is 2^4 = 16.
Common mistake
A negative exponent makes the answer negative.
- Why it's tempting
- A minus sign almost everywhere else in math means a negative number.
- Do this instead
- A negative exponent flips the base into a fraction: 3^(−2) = 1/9. The value is still positive when the base is positive.
Step 4 of 5
What to remember
Remember
- Same base: add exponents to multiply, subtract to divide, multiply for a power of a power.
- A negative exponent means a reciprocal, and a^0 = 1.
- In scientific notation, divide the front numbers and subtract the exponents; line up place values before adding.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, which rule did you use to combine the two powers of 2, and what does the negative exponent mean?