Problem: Simplify each expression.
The task involves simplifying expressions using the rules of exponents. Here are the steps to simplify each expression:
---
####
1. \( x^3 \cdot x^2 \)
-
Rule: When multiplying powers with the same base, add the exponents.
\[
x^a \cdot x^b = x^{a+b}
\]
-
Solution:
\[
x^3 \cdot x^2 = x^{3+2} = x^5
\]
####
2. \( y^0 \)
-
Rule: Any non-zero number raised to the power of 0 is 1.
\[
a^0 = 1 \quad \text{(for } a \neq 0\text{)}
\]
-
Solution:
\[
y^0 = 1
\]
####
3. \( z^4 \cdot z^7 \)
-
Rule: When multiplying powers with the same base, add the exponents.
\[
z^a \cdot z^b = z^{a+b}
\]
-
Solution:
\[
z^4 \cdot z^7 = z^{4+7} = z^{11}
\]
####
4. \( (x^3)(y^3) \)
-
Rule: When multiplying terms with different bases, keep the bases separate and multiply their coefficients if applicable.
\[
(x^a)(y^b) = x^a \cdot y^b
\]
-
Solution:
\[
(x^3)(y^3) = x^3 \cdot y^3
\]
####
5. \( y^2 \cdot y^3 \cdot y^4 \)
-
Rule: When multiplying powers with the same base, add the exponents.
\[
y^a \cdot y^b \cdot y^c = y^{a+b+c}
\]
-
Solution:
\[
y^2 \cdot y^3 \cdot y^4 = y^{2+3+4} = y^9
\]
####
6. \( -x^2 \cdot x^3 \)
-
Rule: When multiplying powers with the same base, add the exponents. The negative sign remains as part of the coefficient.
\[
-x^a \cdot x^b = -x^{a+b}
\]
-
Solution:
\[
-x^2 \cdot x^3 = -x^{2+3} = -x^5
\]
####
7. \( (x^2)^3 \)
-
Rule: When raising a power to another power, multiply the exponents.
\[
(a^m)^n = a^{m \cdot n}
\]
-
Solution:
\[
(x^2)^3 = x^{2 \cdot 3} = x^6
\]
####
8. \( (2x)^3 \)
-
Rule: When raising a product to a power, raise each factor to that power.
\[
(ab)^n = a^n \cdot b^n
\]
-
Solution:
\[
(2x)^3 = 2^3 \cdot x^3 = 8x^3
\]
####
9. \( (xy)^2 \)
-
Rule: When raising a product to a power, raise each factor to that power.
\[
(ab)^n = a^n \cdot b^n
\]
-
Solution:
\[
(xy)^2 = x^2 \cdot y^2
\]
####
10. \( (-3x)^2 \)
-
Rule: When raising a product to a power, raise each factor to that power. Note that \((-3)^2 = 9\).
\[
(ab)^n = a^n \cdot b^n
\]
-
Solution:
\[
(-3x)^2 = (-3)^2 \cdot x^2 = 9x^2
\]
####
11. \( (3x)^2 \)
-
Rule: When raising a product to a power, raise each factor to that power.
\[
(ab)^n = a^n \cdot b^n
\]
-
Solution:
\[
(3x)^2 = 3^2 \cdot x^2 = 9x^2
\]
---
Final Answers:
\[
\boxed{
\begin{array}{lll}
1. & x^5 & \\
2. & 1 & \\
3. & z^{11} & \\
4. & x^3y^3 & \\
5. & y^9 & \\
6. & -x^5 & \\
7. & x^6 & \\
8. & 8x^3 & \\
9. & x^2y^2 & \\
10. & 9x^2 & \\
11. & 9x^2 &
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of multiplying with exponents worksheet.
