Problem 60 Simplify. Write answers in expon... [FREE SOLUTION]

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Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 8: Problem 60

Simplify. Write answers in exponential form with only positive exponents. Assume that all variables represent positive numbers. $$ \sqrt[9]{8^{3}} $$

Short Answer

Expert verified

8^{1/3}

Step by step solution

Understand the nth Root in Exponential Form

To simplify the expression $\root[9]{8^3}$, recognize that nth roots can be expressed as fractional exponents. In this case, $\root[9]{a}$ is the same as $a^{1/9}$.

Apply the Power of a Power Rule

Rewrite the given expression using the fractional exponent rule: $8^3$ under the 9th root becomes $8^{3 \times (1/9)}$.

Simplify the Exponent

Multiply the exponents together: $3 \times (1/9) = 3/9$. Then, simplify $3/9$ to $1/3$.

Write the Final Simplified Expression

The final step is to write the base and the simplified exponent in exponential form. Therefore, $\root[9]{8^3} = 8^{1/3}$.

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Key Concepts

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

fractional exponents

A fractional exponent is a way to express roots in the form of exponents. Think of an exponent as the number of times you multiply a base by itself. Fractional exponents split this concept into both multiplication and roots.
For instance, the expression $a^{1/2}$ is another way to write $\sqrt{a}$ or the square root of 'a'.
The denominator of the fractional exponent tells you the type of root. For example:

$a^{1/3}$ represents the cube root of 'a'.
$a^{1/4}$ represents the fourth root of 'a'.
Similarly, $a^{1/n}$ means the nth root of 'a'.

exponential form

Exponential form is an efficient way to write repetitive multiplication of the same number. Instead of writing $8 \cdot 8 \cdot 8$, you can write $8^3$.
This concept simplifies the notation and makes it easier to handle large numbers.
In our example, we are given $\sqrt[9]{8^3}$. To understand it better, we first convert the root notation into exponential form, which can be written as $8^{3 \cdot (1/9)}$.
This process shows the flexibility of using exponentiation to represent complex algebraic expressions in a compact form.
By doing this, it becomes easier to apply mathematical operations like multiplication and division to exponents.

power of a power rule

The power of a power rule is a key rule when working with exponents. It states that $(a^m)^n = a^{m\cdot n}$.
This means when you raise a power to another power, you multiply the exponents together. For example:

$(x^2)^3 = x^{2\cdot 3} = x^6$
Similarly, $(8^3)^{1/9} = 8^{3/9}$

In our specific problem, $\sqrt[9]{8^3}$ becomes $8^{3 \cdot (1/9)}$. By multiplying the exponents, we simplify it to $8^{1/3}$.
Understanding the power of a power rule helps you manage and simplify expressions involving multiple layers of exponents.

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91影视

Simplify. Write answers in exponential form with only positive exponents. Assume that all variables represent positive numbers. $$ \sqrt[9]{8^{3}} $$

Short Answer

Step by step solution

Understand the nth Root in Exponential Form

Apply the Power of a Power Rule

Simplify the Exponent

Write the Final Simplified Expression

Key Concepts

fractional exponents

exponential form

power of a power rule

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