Problem 44 Rewrite the radical expression i... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

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Chapter 8: Problem 44

Rewrite the radical expression in exponential notation. $$\sqrt{y^{5}}$$

Short Answer

Expert verified

The expression $\sqrt{y^{5}}\rightarrow y^{5/2}$.

Step by step solution

Understand the Radical Expression

A radical can be rewritten using fractional exponents. For instance, the square root of a number can be expressed as raising the number to the power of 1/2.

Convert the Radical to Exponential Form

The given expression is $\root{y^{5}}$. To convert this to exponential form, replace the radical with an exponent of 1/2: $\root{y^{5}} = \root{y^{5}}^{1/2}$.

Apply the Exponential Rule

According to the power rule $(a^{m})^{n} = a^{m \times n}$, simplify the expression: $\root{y^{5}}^{1/2} = y^{5 \times 1/2} = y^{5/2}$.

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Key Concepts

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

fractional exponents

Fractional exponents are a way to represent roots of numbers. For example:
- The square root of a number can be written using a fractional exponent of 1/2.
- Similarly, a cube root is represented as raising a number to the power of 1/3.
In general, the 'n-th' root of a number can be expressed as raising the number to the power of 1/n.
So, $\root{x} = x^{1/2}$ and $\root[3]{x} = x^{1/3}$.
This rule helps simplify complex radical expressions, making them easier to handle in algebra and other math operations.

exponential notation

Exponential notation is a way to express repeated multiplication of a number.
Instead of writing 5 multiplied by itself 3 times (5 * 5 * 5), we write it as $5^{3}$.
The base (5 in this case) is the number being multiplied, and the exponent (3) indicates how many times the base is used as a factor.
Exponential notation makes dealing with large numbers or repeated multiplications simpler and more concise.
When working with radicals, converting them to exponential form can help in simplifying and solving problems more efficiently. For instance: $\sqrt{y^{5}} = y^{5/2}$. This conversion uses the idea that a radical is just an exponent in disguise.

power rule in algebra

The power rule in algebra is fundamental for simplifying expressions involving exponents.
This rule states that $(a^{m})^{n} = a^{m \times n}$.
It tells us what happens when we raise an exponent to another exponent.
For instance, if we have $y^{5}$ and we then take the square root of it, it becomes $ (y^{5})^{1/2} $.
Using the power rule, we multiply the exponents: \ {5 \times 1/2} = 5/2 \. So, \ \sqrt{y^{5}} = y^{5/2} \
This rule makes it easier to manipulate and simplify expressions involving multiple exponents, leading to more straightforward solutions in algebra.

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91影视

Rewrite the radical expression in exponential notation. $$\sqrt{y^{5}}$$

Short Answer

Step by step solution

Understand the Radical Expression

Convert the Radical to Exponential Form

Apply the Exponential Rule

Key Concepts

fractional exponents

exponential notation

power rule in algebra

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