Let’s solve each problem step by step using the
properties of exponents.
We’ll use these key rules:
- When multiplying powers with the same base,
add the exponents:
\( a^m \cdot a^n = a^{m+n} \)
- When dividing powers with the same base,
subtract the exponents:
\( \frac{a^m}{a^n} = a^{m-n} \)
- A negative exponent means “flip” the base to the denominator:
\( a^{-n} = \frac{1}{a^n} \)
- Any nonzero number to the power 0 is 1:
\( a^0 = 1 \) (as long as \( a \ne 0 \))
- Power raised to another power? Multiply the exponents:
\( (a^m)^n = a^{m \cdot n} \)
---
Now let’s go one by one:
1. \( w^3 \cdot w^4 \)
Same base → add exponents:
\( 3 + 4 = 7 \) → \( w^7 \)
2. \( 6x^5 \div 3x^2 \)
First, divide coefficients: \( 6 ÷ 3 = 2 \)
Then subtract exponents for x: \( 5 - 2 = 3 \) → \( 2x^3 \)
3. \( b^8 \cdot b^{-3} \)
Add exponents: \( 8 + (-3) = 5 \) → \( b^5 \)
4. \( \frac{y^9}{y^4} \)
Subtract exponents: \( 9 - 4 = 5 \) → \( y^5 \)
5. \( m^{-2} \cdot m^5 \)
Add exponents: \( -2 + 5 = 3 \) → \( m^3 \)
6. \( \frac{p^7}{p^{-1}} \)
Subtract exponents: \( 7 - (-1) = 7 + 1 = 8 \) → \( p^8 \)
7. \( k^0 \)
Any nonzero base to the 0 power is 1 → \( 1 \)
8. \( z^{-4} \)
Negative exponent → flip to denominator: \( \frac{1}{z^4} \)
9. \( r^6 \cdot r^{-6} \)
Add exponents: \( 6 + (-6) = 0 \) → \( r^0 = 1 \)
10. \( \frac{t^3}{t^3} \)
Subtract exponents: \( 3 - 3 = 0 \) → \( t^0 = 1 \)
11. \( s^{-5} \cdot s^2 \)
Add exponents: \( -5 + 2 = -3 \) → \( s^{-3} = \frac{1}{s^3} \)
12. \( q^{-2} \)
Flip to denominator: \( \frac{1}{q^2} \)
---
✔ All answers checked and verified.
Final Answer:
1. \( w^7 \)
2. \( 2x^3 \)
3. \( b^5 \)
4. \( y^5 \)
5. \( m^3 \)
6. \( p^8 \)
7. \( 1 \)
8. \( \frac{1}{z^4} \)
9. \( 1 \)
10. \( 1 \)
11. \( \frac{1}{s^3} \)
12. \( \frac{1}{q^2} \)
Parent Tip: Review the logic above to help your child master the concept of kuta software simplifying radical expressions.
