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Chapter 10: Problem 528

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

Short Answer

Expert verified
The simplified expression is \(y^{12}\).

Step by step solution

01

Understand the Power Rule

The power rule states that when raising a power to another power, we multiply the exponents. Mathematically, this is expressed as \((a^m)^n = a^{m*n}\).
02

Apply the Power Rule

Identify the base and the exponents in the expression \((y^4)^3\). Here, the base is \(y\), the first exponent is \(4\), and the second exponent is \(3\). According to the power rule, multiply the exponents: \(4 * 3 = 12\).
03

Write the Simplified Expression

Using the result from the multiplication, rewrite the expression with the new exponent: \(y^{4*3} = y^{12}\).

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Key Concepts

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

Exponentiation
Exponentiation is an essential concept in mathematics. It's the process of raising a number or variable to a power, where the power indicates how many times the base is multiplied by itself. For instance, in the expression \(y^4\), \(y\) is the base, and \(4\) is the exponent. This can be read as 'y raised to the fourth power' or 'y to the power of 4'.
When two powers are raised together as in \((y^4)^3\), the exponents multiply, giving the base a new and higher power. For example, the expression \((y^4)^3\) simplifies to \(y^{4*3} = y^{12}\). The Power Rule, which states that \((a^m)^n = a^{m*n}\), is used here to simplify such expressions effectively.
Simplifying Expressions
Simplifying expressions is a vital skill in algebra. It makes complex expressions easier and more manageable. In our example, \((y^4)^3\), using the power rule simplifies it to a more straightforward form much faster.
By understanding the rule that allows us to multiply exponents, we turn \(y^{4*3}\) directly into \(y^{12}\). This method ensures our expression is in its simplest form, which is easier to work with in further calculations. Simplification is not just about making things smaller; it鈥檚 about making them clearer and more understandable.
Prealgebra
Prealgebra is a foundational mathematical subject focused on basic algebra concepts. It bridges the gap between arithmetic and algebra, ensuring students are ready for more complicated equations.
One of the key skills in prealgebra is understanding and working with exponents. For instance, in the problem \((y^4)^3\), students learn to apply the power rule which is crucial for their algebraic studies.
Mastering these basics is significant because it builds the groundwork for more advanced topics. By mastering the Power Rule, students can confidently progress to more complex algebraic expressions and equations.

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Most popular questions from this chapter

Convert each number to decimal form. $$9.413 \times 10^{-5}$$

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Simplify each expression. $$(s+8)(s+9)$$
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