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Chapter 5: Problem 140

Simplify each expression. $$ \left(3 y^{2}\right)^{4} $$

Short Answer

Expert verified
81y^{8}

Step by step solution

01

Understand the power of a power rule

When simplifying expressions with exponents, use the power of a power rule: \ \(\left(a^m \right)^n = a^{mn}\).
02

Apply the power of a power rule

Identify the base and the exponents in the given expression: \ \( \left(3 y^{2} \right)^{4} \). Here, the base is \( 3y^2 \), the inner exponent is 2, and the outer exponent is 4. According to the rule, multiply the exponents: \ \( (y^{2})^{4} = y^{2 \cdot 4} = y^8 \).
03

Apply the power to the coefficient

Now, apply the outer exponent to the coefficient 3 as well. So, \( \left(3 \right)^{4} = 81 \).
04

Combine the results

Combine the simplified parts to get the final result: \ \( 81 y^{8} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of a Power Rule
In mathematics, dealing with exponents can sometimes be tricky, but understanding the power of a power rule makes it a lot easier. The power of a power rule states that when you raise an exponent to another exponent, you multiply the exponents together. For example, \(\big(a^m\big)^n = a^{mn}\). Let's look at our exercise where we have \(\big(3y^2\big)^4\). Here, the exponent 2 is raised to another exponent 4. Using the rule: \(\big(y^2\big)^4 = y^{2 \times 4} = y^8\). This simplifies the expression significantly.
Simplifying Expressions
Simplifying expressions with exponents means breaking them down to their simplest form. In our example, we dealt with \(\big(3y^2\big)^4\). By simplifying, we follow these steps: first, apply the power of a power rule to the variable part, i.e., \(\big(y^2\big)^4 = y^{2 \times 4} = y^8\). Next, we need to handle the coefficient, which is the number part. So, \(3\) raised to the power of 4 becomes \(\big(3\big)^4 = 81\). Putting these simplified parts together, we get \(81 y^8\). Breaking the problem into smaller steps makes it easier to manage and understand.
Exponent Multiplication
Another key concept in simplifying expressions is exponent multiplication. This comes in handy when you are dealing with multiple exponents, like in our example \(\big(3y^2\big)^4\). How does it work? When you multiply exponents, you add the powers if the bases are the same and multiply the powers in nested exponents. Let's see the example: \(\big(y^2\big)^4\). According to the power of a power rule, we multiply the exponents, so \(\big(y^2\big)^4 = y^{2 \times 4} = y^8\). This clearly demonstrates how exponent multiplication simplifies the calculation. Combining this with the coefficient part, \(\big(3\big)^4 = 81\), results in our simplified expression \(81 y^8\).

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