Question

In Problems 87-96, simplify each expression. Express the answer so that all exponents are positive. Whenever an exponent is 0 or negative, we assume that the base is not \(0 .\) \(\left(9 x^{4}\right)^{2}\)

Short answer

The simplified expression is \( 81x^8 \).

Step-by-step solution

- Apply the Power Rule

Use the power rule \((a^m)^n = a^{mn}\) to simplify the given expression. Here the expression is \( (9x^4)^2 \). So, apply the power rule to both the constant 9 and the variable term \(x^4\).

- Simplify the constant

Simplify the constant term raised to the power 2. \( 9^2 = 81 \).

- Simplify the variable term

Now apply the power rule to the variable term. \( (x^4)^2 = x^{4 \times 2} = x^8 \).

- Combine the simplified terms

Combine the simplified constant term and the variable term. The simplified expression is \( 81x^8 \).

Key concepts

power rule

When working with algebraic expressions that involve exponents, the power rule is a useful tool. Follow these steps to understand and apply the power rule:
The power rule states: \((a^m)^n = a^{mn}\). This means if you raise an exponent to another exponent, you multiply the exponents together.
For example, to simplify \((9x^4)^2\):

• First, apply the power rule to the constant 9 and the variable term \(x^4\).
• The constant 9 raised to the power of 2 is calculated as \(9^2 = 81\).
• For the variable term, apply the power rule to get \((x^4)^2 = x^{4 \times 2} = x^8\).

Combining these results, we obtain a simplified expression of \(81x^8\).

simplifying expressions

Simplifying an expression means reducing it to its most basic form. This often involves:

• Combining like terms
• Applying mathematical rules (such as the power rule)
• Ensuring no negative or zero exponents

Let's revisit our example: \((9x^4)^2\). By applying the power rule, we broke down the expression into simpler parts. First, we dealt with the constant and the variable separately. This approach makes it easier to handle complex expressions step by step. Once each part is simplified, we combine them back to form a neat, final expression. In this case, the result was \(81x^8\). This method ensures clarity and accuracy in solving algebraic problems.

exponents

Exponents are numbers that indicate how many times a base is multiplied by itself. They are a fundamental part of algebra and appear frequently in expressions:

• An exponent of 2 means the base is squared, such as \(a^2 = a \times a\)
• An exponent of 3 indicates cubing the base, \(a^3 = a \times a \times a\)

In our exercise, we simplified \((9x^4)^2\) where the exponent 2 applies to both the constant (9) and the exponent (4). Knowing how to manipulate exponents using rules like the power rule is essential in algebra. This helps in breaking down and simplifying more complex expressions, leading to clearer and accurate final results, just like we achieved with \(81x^8\).