Problem 136 Simplify each expression. $$ ... [FREE SOLUTION]

Chapter 5: Problem 136

Simplify each expression. $$ \left(y^{4}\right)^{3} \cdot\left(y^{5}\right)^{2} $$

Short Answer

Expert verified

The simplified expression is $y^{22}$.

Step by step solution

Apply the Power of a Power Rule

For the term $\big(y^{4}\big)^{3}$, apply the power of a power rule, which states that $ (a^m)^n = a^{m \times n} $. So, $\big(y^{4}\big)^{3} = y^{4 \times 3} = y^{12}$. Similarly, for $\big(y^{5}\big)^{2}$, apply the same rule to get $\big(y^{5}\big)^{2} = y^{5 \times 2} = y^{10}$.

Combine the Exponents Using the Product Rule

Now that both terms have been simplified, multiply them together using the product rule for exponents, which states that $a^m \times a^n = a^{m+n}$ for the same base. Here, \y^{12} \times y^{10}\ can be combined as \y^{12+10} = y^{22}\.

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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

The power of a power rule is a key concept in simplifying exponents. When you see an expression like $(y^4)^3$, it means you are raising an exponent to another exponent. According to the power of a power rule, $ (a^m)^n = a^{mn}$.

To apply this to $(y^4)^3$, you multiply the exponents together: $y^{4 \times 3} = y^{12}$.

Another example is $(y^5)^2$. Using the same rule, you get $y^{5 \times 2} = y^{10}$. This method makes it easy to simplify even complex exponent expressions.

Remember, this rule only works when the base is the same, and you're raising one exponent to another.

product rule for exponents

The product rule for exponents is especially useful when combining terms with the same base. It states that $a^m \times a^n = a^{m+n}$.

Let's apply this rule to our simplified expression: $y^{12} \times y^{10}$. Both terms have the same base 'y', so you can add the exponents together: $y^{12+10} = y^{22}$.

Using the product rule helps streamline calculations and simplifies otherwise cumbersome multiplications.

algebraic expressions

Understanding algebraic expressions involving exponents is crucial. An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols.

In the given exercise, $ (y^4)^3 \times (y^5)^2 $, the expression consists of individual exponent terms that can be simplified using the power of a power rule and then combined using the product rule.

Remember, the key aspects when working with algebraic expressions involving exponents are:

Identifying the base and the exponents
Applying the correct exponent rules
Combining like terms appropriately

This gradual approach makes complex expressions easier to handle.

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91影视

Simplify each expression. $$ \left(y^{4}\right)^{3} \cdot\left(y^{5}\right)^{2} $$

Short Answer

Step by step solution

Apply the Power of a Power Rule

Combine the Exponents Using the Product Rule

Key Concepts

power of a power rule

product rule for exponents

algebraic expressions

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