Problem 55 Simplify the expression. $$\le... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 10: Problem 55

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

Short Answer

Expert verified

$y^{14}$

Step by step solution

- Apply the Power Rule

When raising a power to another power, multiply the exponents. For $\left(y^{2}\right)^{3}$, use the rule $\left(a^{m}\right)^{n} = a^{m \cdot n}$. This gives $\left(y^{2}\right)^{3} = y^{2 \cdot 3} = y^{6}$.

- Simplify Each Group Separately

Now apply the same rule to the second part of the expression $\left(y^{4}\right)^{2}$. Again, use $\left(a^{m}\right)^{n} = a^{m \cdot n}$. This gives $\left(y^{4}\right)^{2} = y^{4 \cdot 2} = y^{8}$.

- Multiply the Simplified Expressions

Now multiply the two simplified expressions together. This uses the rule $a^{m} \cdot a^{n} = a^{m+n}$. So, $y^{6} \cdot y^{8} = y^{6+8} = y^{14}$.

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Key Concepts

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

Power Rule

The power rule in algebra is a fundamental tool that simplifies working with exponents. When you raise a power to another power, you multiply the exponents. This rule can be expressed as $(a^m)^n = a^{m \times n}$. For instance, if you have $ (y^2)^3 $, you multiply the exponents 2 and 3, resulting in $ y^{2 \times 3} = y^6 $. This technique is crucial for handling more complex expressions involving exponents, making calculations more straightforward.

Multiplying Exponents

Multiplying exponents becomes easy once you understand the underlying rules. When you have expressions with the same base, you can simply add the exponents. The rule for this is $a^m \times a^n = a^{m+n}$. Applying this to the provided problem, once you've simplified using the power rule, you end up with $ y^6 \times y^8 $. Add the exponents 6 and 8, which gives $ y^{6+8} = y^{14} $. Understanding this rule helps in quickly reducing expressions to their simplest form.

Algebra Rules

Algebra involves several fundamental rules that help in simplifying expressions. The two key rules used in this exercise are:

Power Rule: $ (a^m)^n = a^{m \times n} $
Multiplying Exponents: $ a^m \times a^n = a^{m+n} $

By applying these rules step-by-step, you can break down complex expressions into simpler parts for easier manipulation and calculation. In the given exercise, following these algebra rules allowed us to simplify $ (y^2)^3 (y^4)^2 $ efficiently to $ y^{14} $. Mastering these rules forms a strong foundation for solving various algebraic problems.

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91影视

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

Short Answer

Step by step solution

- Apply the Power Rule

- Simplify Each Group Separately

- Multiply the Simplified Expressions

Key Concepts

Power Rule

Multiplying Exponents

Algebra Rules

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