Question

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$\left(x^{4} y^{8}\right)^{3 / 4}$$

Short answer

x^3 y^6

Step-by-step solution

- Apply the power rule for exponents

Use the power rule \((a^m)^n = a^{m \times n}\) to simplify the expression. Here, take each exponent inside the parenthesis and multiply it by \(3/4\). This gives us: \[\left(x^{4} y^{8}\right)^{3/4} = x^{4 \times (3/4)} y^{8 \times (3/4)}\].

- Simplify the exponents

Now, simplify each of the exponents: \4 \times (3/4) = 3\ and \8 \times (3/4) = 6\. This results in: \[x^3 y^6\].

Key concepts

Power Rule for Exponents

To simplify expressions like \((x^4 y^8)^{3/4}\), we use the power rule for exponents. The power rule states that when raising a power to another power, you multiply the exponents: \((a^m)^n = a^{mn}\). This rule is very helpful in breaking down complex expressions.
In our example, we can apply this rule to both \(x^4\) and \(y^8\). We do this by taking each exponent inside the parenthesis and multiplying it by \(3/4\). So, the expression becomes:

• \(x^{4 \times (3/4)}\)
• \(y^{8 \times (3/4)}\)

This results in:

• \(x^3\)
• \(y^6\)

Now, the expression \((x^4 y^8)^{3/4}\) has been simplified to:\(x^3 y^6\).

Positive Exponents

Expressions involving exponents are often simplified to ensure they contain only positive exponents. Positive exponents indicate that the base is being multiplied by itself a certain number of times. For example, \(x^3\) means \(x \times x \times x\).
It’s important to express answers with positive exponents for clarity and to adhere to standard mathematical conventions. In our example, the initial expression is \( (x^4 y^8)^{3/4} \). By applying the power rule for exponents, we end up with \( x^3 y^6 \), where both exponents (3 and 6) are positive. This makes it easier to understand and consistent with mathematical norms.

Multiplication of Exponents

Multiplying exponents is a crucial part of simplifying expressions, especially when using the power rule. When multiplying exponents, ensure that you're multiplying the exponents outside and inside the parenthesis correctly.
Let's look at our example: \((x^4 y^8)^{3/4}\). According to the power rule, we multiply each exponent by \(3/4\):

• For \(x^4\), we have \(4 \times (3/4) = 3\)
• For \(y^8\), we have \(8 \times (3/4) = 6\)

This step-by-step multiplication gives us the simplified exponents:

• \(x^3\)
• \(y^6\)

Thus, the resulting simplified expression is \(x^3 y^6\). By closely following the multiplication of exponents, we ensure the accuracy of our solution.