Part 1: Evaluating Expressions

1. Evaluate: $(-2)^5$

2. Find the value of: $(-4)^2$

3. Simplify: $(-3)^6$

4. Evaluate: $(-7)^3$

5. Find the value of: $(-5)^4$

Part 2: Solving for $x$

6. If $2^{x+4} = 64$, find $x$.

7. If $3^x = 81$, find $x$.

8. If $5^{x-1} = 25$, find $x$.

9. If $4^x = 16$, find $x$.

10. If $2^{2x} = 32$, find $x$.

Part 1: Evaluating Negative Bases

When dealing with negative numbers and exponents, remember: if the exponent is even, the result is positive. If the exponent is odd, the result is negative.

1. Evaluate: $(-2)^5$

   • Explanation: This means multiplying $-2$ by itself five times: $(-2) \times (-2) \times (-2) \times (-2) \times (-2)$. Since 5 is an odd number, the result stays negative.

   • Result: $-32$

2. Find the value of: $(-4)^2$

   • Explanation: This is $(-4) \times (-4)$. Since a negative times a negative is a positive (and 2 is an even exponent), the result is positive.

   • Result: $16$

3. Simplify: $(-3)^6$

   • Explanation: Multiply $-3$ by itself six times. Because 6 is even, the answer is positive. $3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729$.

   • Result: $729$

4. Evaluate: $(-7)^3$

   • Explanation: Multiply $-7$ by itself three times. Since 3 is odd, the result is negative. $7 \times 7 \times 7 = 343$.

   • Result: $-343$

5. Find the value of: $(-5)^4$

   • Explanation: Multiply $-5$ by itself four times. Since 4 is even, the result is positive. $5 \times 5 \times 5 \times 5 = 625$.

   • Result: $625$

Part 2: Solving Exponential Equations

To solve these, write both sides of the equation with the same base, then set the exponents equal to each other.

6. If $2^{x+4} = 64$, find $x$.

   • Explanation: First, write 64 as a power of 2: $64 = 2^6$. Now the equation is $2^{x+4} = 2^6$. Since the bases are the same, $x + 4 = 6$.

   • Result: $x = 2$

7. If $3^x = 81$, find $x$.

   • Explanation: Write 81 as a power of 3: $81 = 3 \times 3 \times 3 \times 3$, which is $3^4$. So, $3^x = 3^4$.

   • Result: $x = 4$

8. If $5^{x-1} = 25$, find $x$.

   • Explanation: Write 25 as $5^2$. The equation becomes $5^{x-1} = 5^2$. Set the exponents equal: $x – 1 = 2$.

   • Result: $x = 3$

9. If $4^x = 16$, find $x$.

   • Explanation: Write 16 as $4^2$. The equation is $4^x = 4^2$.

   • Result: $x = 2$

10. If $2^{2x} = 32$, find $x$.

    • Explanation: Write 32 as a power of 2: $32 = 2^5$. The equation is $2^{2x} = 2^5$. Set the exponents equal: $2x = 5$.

    • Result: $x = 2.5$ (or $5/2$)