Question

Write your answer as a power or as a product of powers. $$ 5^{3} \cdot\left(5 a^{4}\right)^{2} $$

Short answer

The answer is \(5^{5} * a^{8}\)

Step-by-step solution

Apply Powers-of-Products Rule

We start by applying the powers-of-products rule to \((5a^{4})^{2}\), which says that the power applies to both the numeric and variable parts of the product. This gives us \(5^{2} * a^{4*2}\) which simplifies to \(5^{2} * a^{8}\).

Apply Product-to-Powers Rule

The next step is to apply the product-to-powers rule to the entire expression. This allows us to add the exponents of like bases together. We have the \(5^{3}\) and the \(5^{2}\) that we just derived in the previous step. Adding those together gets us \(5^{3+2}\) which simplifies to \(5^{5}\).

Combine Like Terms

The final step is to combine the \(5^{5}\) expression with the \(a^{8}\) from step 1, yielding the final result.

Key concepts

Understanding Exponents

Exponents, also referred to as powers, are a shorthand notation in math that tells us how many times to multiply a number by itself. For instance, the expression \( 5^3 \) means that the number 5 is to be multiplied by itself three times, which is \( 5 \times 5 \times 5 = 125 \). It's important to know that an exponent applies only to the number it is directly attached to. When dealing with exponents, there are specific rules designed to simplify expressions, one such rule being the power-of-a-power rule.

Simplifying expressions with exponents correctly requires attention to detail and understanding how exponents interact with each other and with different operations such as multiplication and division. By mastering these rules, you make the process of working with algebraic expressions much more straightforward.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a foundational skill in algebra. It involves combining like terms and using algebraic properties to make expressions easier to work with. In our example, we simplify the expression \( (5a^{4})^{2} \) by recognizing that both the coefficient (5) and the variable part (\( a^{4} \)) are raised to the second power. This means we apply the power to both, turning it into \( 5^2 \) and \( a^{8} \).

Being able to simplify expressions not only makes them neater but also prepares them for further operations, such as solving equations. It's a crucial step and, as such, understanding the order of operations and the relevant algebra rules, such as the distributive property, is essential.

Multiplying Powers with the Same Base

When you come across expressions that involve multiplying powers with the same base, you can simplify the expression by adding the exponents. This rule applies because you're effectively multiplying out all the factors involved; for instance, \( 5^3 \times 5^2 \) is the same as \( 5 \times 5 \times 5 \times 5 \times 5 \), which we know is \( 5^5 \).

This rule is a part of the properties of exponents that greatly simplifies dealing with them, essentially boiling down to keeping the base and adding the exponents. This simplification can only be applied when the bases are the same and is particularly useful when dealing with algebraic expressions that include variables.