Exponent Rules - Pre Algebra

1. What is the product rule for exponents with the same base?

   $a^m \cdot a^n = a^{m+n}$. Keep the base and add the exponents.

2. Simplify $x^2 \cdot x^6$.

   $x^8$. Add the exponents of the common base.

3. What is the quotient rule for exponents with the same base?

   $\frac{a^m}{a^n} = a^{m-n}$ for $a \ne 0$. Keep the base and subtract the exponents.

4. Why does the product rule not combine $2^3 \cdot 5^4$ into one power?

   The bases differ. The product rule $a^m \cdot a^n = a^{m+n}$ applies only when the bases are the same.

5. What is the error in $4^2 \cdot 4^3 = 16^5$?

   The bases were multiplied. The base stays the same, so $4^2 \cdot 4^3 = 4^5$.

6. What is the power of a power rule?

   $(a^m)^n = a^{mn}$. Keep the base and multiply the exponents.

7. Simplify $3^4 \cdot 3^5$ as a single power.

   $3^9$. Add the exponents: $4 + 5 = 9$.

8. Simplify $(3x)^2$.

   $9x^2$. Square each factor: $3^2 \cdot x^2$.

9. Simplify $\frac{7^8}{7^3}$ as a single power.

   $7^5$. Subtract the exponents: $8 - 3 = 5$.

10. What is the power of a product rule?

    $(ab)^n = a^n b^n$. Raise each factor to the power.

11. Simplify $\frac{5^6 \cdot 5^2}{5^4}$ as a single power.

    $5^4$. The numerator is $5^8$, and $8 - 4 = 4$.

12. Simplify $(2^3)^4$ as a single power.

    $2^{12}$. Multiply the exponents: $3 \cdot 4 = 12$.

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