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The product-to-powers rule is an essential rule when working with exponents in algebra. This rule states that when you have a power outside a parenthesis that contains a product (several terms multiplied within it), you can "distribute" the exponent to each factor inside the parentheses. For example, in the expression \((ab)^n\), apply the power \(n\) to both \(a\) and \(b\), which results in

• \((ab)^n = a^n \cdot b^n\)

This rule allows you to separate complex expressions into simpler ones by treating each factor individually.

In the exercise \((-2 v^{5})^{4}\), the base includes two factors,

• -2 and
• \(v^{5}\)

Apply the exponent 4 to both:

• \((-2)^4\) and
• \((v^{5})^4\).

This is the power that each part gets when you simplify the expression later on.

Simplifying expressions with exponents involves breaking them down to their most basic form. This often requires using both the product-to-powers rule, as well as understanding how to multiply and divide powers. When you simplify

• expressions like \((-2)^4\) and
• \((v^{5})^4\),

you work on each part separately and then combine the results.

For the expression \((-2)^4\), repeatedly multiplying

• -2 by itself results in 16, because multiplying an even number of negatives gives a positive.

Therefore, \((-2)^4 = 16\).

For the expression \((v^{5})^4\), apply the rule \(a^{m\times n} = a^{m\cdot n}\), which means you multiply the exponents to get \(v^{20}\).

Combining these results, you get the simplified expression:

• \(16 \cdot v^{20}\).

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators (like +, −, ×, ÷). In algebra, you often work on expressions to simplify or solve them, just like in our exercise. Simplifying expressions with exponents, such as \((-2 v^{5})^{4}\), helps reduce complex statements into easier forms.

• Variables (\(v\)) represent unknown values and can have exponents.
• Constants (like -2) are numbers on their own.
• Operations (such as powers) indicate relationships amongst constants and variables.

When working with expressions containing both constants and variables, understanding the rules of exponents is crucial.

These rules, including the product-to-powers rule, enable you to rewrite and condense expressions into their simplest form, making it easier to work with them effectively.

This simplification is the backbone of solving algebraic problems, allowing deeper insights into more complex algebraic tasks.