Use the rules of exponents to simplify the expressions. Match the expression with its equivalent value.

Question

1. (-2)exponent-5/ (-2)exponent-10

2. 2 exponent-1 ∙ 2 exponent-4
-(-1/2)exponent3 * (-1/2)exponent2
2/ 2exponent-4
3. ∙
1/32
4.
-1/32

Answer 1

To simplify the given expressions using the rules of exponents, let's analyze each one:

1. Expression: $(-2{)}^{-5}/(-2{)}^{-10}$

Using the rule $\frac{a^{-m}}{a^{-n}}=a^{n-m}$:

$$(-2{)}^{-5}/(-2{)}^{-10}=(-2{)}^{-5-(-10)}=(-2{)}^{-5+10}=(-2{)}^5$$

Since $(-2{)}^5=-32$, the value is:

$$-32$$

2. Expression: $2^{-1}\cdot 2^4$

Using the rule $a^m\cdot a^n=a^{m+n}$:

$$2^{-1}\cdot 2^4=2^{-1+4}=2^3$$

Since $2^3=8$, the value is:

$$8$$

3. Expression: $-(-\frac{1}{2}{)}^3\cdot (-\frac{1}{2}{)}^2$

Using the rule $a^m\cdot a^n=a^{m+n}$:

$$-(-\frac{1}{2}{)}^3\cdot (-\frac{1}{2}{)}^2=-(-\frac{1}{2}{)}^{3+2}=-(-\frac{1}{2}{)}^5$$

Calculating $(-1/2{)}^5$:

$$(-\frac{1}{2}{)}^5=-\frac{1}{32}$$ So:

$$-(-\frac{1}{32})=\frac{1}{32}$$

4. Expression: $\frac{2}{2^{-4}}$

Using the rule $\frac{a}{a^{-n}}=a^{1+n}$:

$$\frac{2}{2^{-4}}=2^{1-(-4)}=2^{1+4}=2^5$$

Since $2^5=32$, the value is:

$$32$$

Final Matches:

• $(-2{)}^{-5}/(-2{)}^{-10}\to -32$
• $2^{-1}\cdot 2^4\to 8$
• $-(-\frac{1}{2}{)}^3\cdot (-\frac{1}{2}{)}^2\to \frac{1}{32}$
• $\frac{2}{2^{-4}}\to 32$

Thus, matching up:

1. $(-2{)}^{-5}/(-2{)}^{-10}\to -32$
2. $2^{-1}\cdot 2^4\to 8$
3. $-(-\frac{1}{2}{)}^3\cdot (-\frac{1}{2}{)}^2\to \frac{1}{32}$
4. $\frac{2}{2^{-4}}\to 32$

If you have specific expressions you need matched or corrected, please clarify!