Answer
1. \( 2^{6} \times 2^{-3} = 8 \)
2. \( \left(-2 a^{2}\right)^{5} = -32a^{10} \)
3. \( \frac{16 x^{-2} y^{2}}{32 x y^{-2}} = \frac{y^{4}}{2x^{3}} \)
4. \( \left(\frac{4 a b^{-3}}{12 a^{-1} b^{2}}\right)^{-3} = \frac{27 b^{15}}{a^{6}} \)
Solution
Let's simplify each expression step by step.
### 4.1. \( 2^{6} \times 2^{-3} \)
**Solution:**
Use the property of exponents \( a^m \times a^n = a^{m+n} \).
\[
2^{6} \times 2^{-3} = 2^{6-3} = 2^{3} = 8
\]
**Simplified Form:**
\[
8
\]
---
### 4.2. \( \left(-2 a^{2}\right)^{5} \)
**Solution:**
Apply the exponent to both the coefficient and the variable:
\[
(-2)^5 \times (a^{2})^5 = -32 \times a^{10} = -32a^{10}
\]
**Simplified Form:**
\[
-32a^{10}
\]
---
### 4.3. \( \frac{16 x^{-2} y^{2}}{32 x y^{-2}} \)
**Solution:**
Simplify the coefficients and apply the properties of exponents:
1. **Coefficients:**
\[
\frac{16}{32} = \frac{1}{2}
\]
2. **Variable \( x \):**
\[
x^{-2} \div x = x^{-2 - 1} = x^{-3}
\]
3. **Variable \( y \):**
\[
y^{2} \div y^{-2} = y^{2 - (-2)} = y^{4}
\]
Combine all parts:
\[
\frac{1}{2} \times x^{-3} \times y^{4} = \frac{y^{4}}{2x^{3}}
\]
**Simplified Form:**
\[
\frac{y^{4}}{2x^{3}}
\]
---
### 4.4. \( \left(\frac{4 a b^{-3}}{12 a^{-1} b^{2}}\right)^{-3} \)
**Solution:**
1. **Simplify the inner fraction:**
- **Coefficients:**
\[
\frac{4}{12} = \frac{1}{3}
\]
- **Variable \( a \):**
\[
\frac{a}{a^{-1}} = a^{1 - (-1)} = a^{2}
\]
- **Variable \( b \):**
\[
\frac{b^{-3}}{b^{2}} = b^{-3 - 2} = b^{-5}
\]
So, the inner fraction becomes:
\[
\frac{1}{3} a^{2} b^{-5}
\]
2. **Apply the exponent \(-3\):**
\[
\left(\frac{1}{3} a^{2} b^{-5}\right)^{-3} = \left(\frac{1}{3}\right)^{-3} \times a^{2 \times (-3)} \times b^{-5 \times (-3)}
\]
\[
= 3^{3} \times a^{-6} \times b^{15} = 27 \times a^{-6} \times b^{15} = \frac{27 b^{15}}{a^{6}}
\]
**Simplified Form:**
\[
\frac{27 b^{15}}{a^{6}}
\]
---
**Summary of Simplified Expressions:**
1. \( 2^{6} \times 2^{-3} = 8 \)
2. \( \left(-2 a^{2}\right)^{5} = -32a^{10} \)
3. \( \frac{16 x^{-2} y^{2}}{32 x y^{-2}} = \frac{y^{4}}{2x^{3}} \)
4. \( \left(\frac{4 a b^{-3}}{12 a^{-1} b^{2}}\right)^{-3} = \frac{27 b^{15}}{a^{6}} \)
Reviewed and approved by the UpStudy tutoring team
Explain
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