Question

example 1 write each product using exponents.
a. \\((-9) \cdot (-9) \cdot (-9) \cdot (-9) \cdot (-9)\\)
because \\(-9\\) is used as a factor 5 times, its exponent is 5.
so, \\((-9) \cdot (-9) \cdot (-9) \cdot (-9) \cdot (-9) = (-9)^5\\).
b. \\(\pi \cdot pi \cdot h \cdot h \cdot h\\)
because \\(\pi\\) is used as a factor 2 times, its exponent is 2. because \\(h\\) is used as a factor 3 times, its exponent is 3.
so, \\(\pi \cdot pi \cdot h \cdot h \cdot h = \pi^2 h^3\\).

example 2 evaluate each expression.
a. \\((-5)^4\\)
\\((-5)^4 = (-5) \cdot (-5) \cdot (-5) \cdot (-5)\\) write as repeated multiplication.
\\(= 625\\) simplify.
b. \\(-5^4\\)
\\(-5^4 = -(5 \cdot 5 \cdot 5 \cdot 5)\\) write as repeated multiplication.
\\(= -625\\) simplify.

Explanation:

Analyze the image content

The image is oriented sideways (rotated 90 degrees counterclockwise) and contains two worked examples demonstrating rules of exponents:

• Example 1: Write each product using exponents.
• a. \((-9) \cdot (-9) \cdot (-9) \cdot (-9) \cdot (-9) = (-9)^5\)
• b. \(\pi \cdot \pi \cdot h \cdot h \cdot h = \pi^2 h^3\)
• Example 2: Evaluate each expression.
• a. \((-5)^4 = (-5) \cdot (-5) \cdot (-5) \cdot (-5) = 625\)
• b. \(-5^4 = -(5 \cdot 5 \cdot 5 \cdot 5) = -625\)

Identify the user's request

The user has uploaded an image containing textbook examples explaining how to write products using exponents and how to evaluate exponential expressions (specifically highlighting the difference between evaluating a negative base raised to a power, \((-5)^4\), versus the negative of a power, \(-5^4\)). Since there are no unsolved exercises or questions presented in the image, this is an explanatory overview of the mathematical concepts shown.

Answer:

The image contains two worked examples explaining the rules of exponents:

Example 1: Write each product using exponents

This example demonstrates how to write repeated multiplication in exponential form \(a^n\), where the base \(a\) is the repeated factor and the exponent \(n\) is the number of times the factor is multiplied.

• a. \((-9) \cdot (-9) \cdot (-9) \cdot (-9) \cdot (-9)\)
• Since the factor \(-9\) is multiplied \(5\) times, the base is \(-9\) and the exponent is \(5\).
• Result: \((-9)^5\)
• b. \(\pi \cdot \pi \cdot h \cdot h \cdot h\)
• The factor \(\pi\) is multiplied \(2\) times (exponent \(2\)), and the factor \(h\) is multiplied \(3\) times (exponent \(3\)).
• Result: \(\pi^2 h^3\)

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Example 2: Evaluate each expression

This example highlights the critical difference between raising a negative number to a power versus applying a negative sign to a power.

• a. \((-5)^4\)
• The parentheses indicate that the base is \(-5\). We multiply \(-5\) by itself \(4\) times.
• \((-5) \cdot (-5) \cdot (-5) \cdot (-5) = 625\)
• Result: \(625\) (an even number of negative factors yields a positive product).
• b. \(-5^4\)
• Without parentheses, the base is only \(5\), and the negative sign is applied to the final result.
• \(-(5 \cdot 5 \cdot 5 \cdot 5) = -625\)
• Result: \(-625\)