Question

Use the exponent properties to simplify each expression.
(x³y⁴) (x²y³)
(5y²w⁴)²(yw²)⁴
y⁹/y⁶
r⁶n-⁷/r⁴n²
(4x²y³)⁰/8x³
15x³y⁵/18x⁵y²
(5xy²z⁴/3xyz)³

Answer 1

Sure, let's simplify each of these expressions step by step using exponent properties.

1. **(x³y⁴)(x²y³)**
- Combine like bases by adding their exponents:
\[
x^{3+2}y^{4+3} = x^5y^7
\]

2. **(5y²w⁴)²(yw²)⁴**
- First simplify each part:
\[
(5^2y^{2 \cdot 2}w^{4 \cdot 2})(y^1w^{2 \cdot 4}) = (25y^4w^8)(y^4w^8)
\]
- Combine like terms:
\[
25y^{4+4}w^{8+8} = 25y^8w^{16}
\]

3. **y⁹/y⁶**
- Use the property of exponents for division:
\[
y^{9-6} = y^3
\]

4. **r⁶n⁻⁷/r⁴n²**
- Divide the bases by subtracting the exponents:
\[
r^{6-4}n^{-7-2} = r^2n^{-9} = \frac{r^2}{n^9}
\]

5. **(4x²y³)⁰/8x³**
- Anything raised to the power of zero is 1 (except when it's zero itself):
\[
\frac{1}{8x^3}
\]

6. **15x³y⁵/18x⁵y²**
- Simplify the coefficients and then apply the exponent properties:
\[
\frac{15}{18} \cdot \frac{x^{3-5}}{y^{5-2}} = \frac{5}{6} \cdot \frac{1}{x^2} \cdot \frac{1}{y^3} = \frac{5}{6x^2y^3}
\]

7. **(5xy²z⁴/3xyz)³**
- First simplify the fraction:
\[
\frac{5}{3} \cdot \frac{x^{1}}{x^{1}} \cdot \frac{y^{2}}{y^{1}} \cdot \frac{z^{4}}{z^{1}} = \frac{5}{3} \cdot 1 \cdot y^{2-1} \cdot z^{4-1} = \frac{5}{3} y^1 z^3 = \frac{5yz^3}{3}
\]
- Now raise the simplified fraction to the power of 3:
\[
\left(\frac{5yz^3}{3}\right)^3 = \frac{5^3(y^3)(z^{9})}{3^3} = \frac{125y^3z^9}{27}
\]

Here are the simplified expressions:
1. \( x^5y^7 \)
2. \( 25y^8w^{16} \)
3. \( y^3 \)
4. \( \frac{r^2}{n^9} \)
5. \( \frac{1}{8x^3} \)
6. \( \frac{5}{6x^2y^3} \)
7. \( \frac{125y^3z^9}{27} \)