Problem 22 Write each expression in exponen... [FREE SOLUTION]

Chapter 7: Problem 22

Write each expression in exponential form. $$\sqrt[3]{a^{2}}$$

Short Answer

Expert verified

The equivalent exponential form of $\sqrt[3]{a^{2}}$ is $a^{2/3}$.

Step by step solution

Converting Radical to Exponential Form

To convert a radical to an exponential form, the formula $\sqrt[n]{a^m} = a^{m/n}$ is used. Here, $\sqrt[3]{a^{2}}$ represents the cube root of $a^2$. Substitute $n=3$ and $m=2$ into the given formula.

Substitution

Plugging the values $n=3$ and $m=2$ into the equivalent exponential form will yield $a^{m/n}=a^{2/3}$.

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Key Concepts

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

Radicals

In mathematics, radicals refer to expressions that involve roots, such as square roots or cube roots. The radical symbol, $\sqrt{}$, is used to denote a root. Radicals help simplify complex expressions by indicating how many times a number must be multiplied by itself to achieve another number. For example, $\sqrt{9}$ means "what number times itself equals 9?" and the answer is 3.

Understanding Radicals: The number under the radical sign is called the radicand, and the small number outside the radical symbol, known as the index, tells us which root to take.
Examples of Radicals: $\sqrt{16} = 4$ and $\sqrt[3]{27} = 3$.

In the expression $\sqrt[3]{a^2}$, "$a^2$" is the radicand, and 3 is the index, which means we are looking for the cube root.

Exponents

Exponents are a mathematical shorthand for expressing repeated multiplication. When you see a number like $a^n$, it means you multiply "a" by itself "n" times. Here, "a" is the base, and "n" is the exponent.
In the expression $a^2$, the base "a" is multiplied by itself once (since 2 - 1 = 1), giving you $a \times a = a^2$.

Exponential Growth: As the exponent increases, the value of the expression can grow very quickly.
Exponent Rules: Common exponent rules include multiplying powers with the same base by adding exponents, and power of a power rule ($ (a^m)^n = a^{m \cdot n} $).

Exponents simplify mathematical expressions and make it easier to perform calculations involving large numbers.

Rational Exponents

Rational exponents are a way to express roots using exponents, where the exponent is a fraction. The expression $a^{m/n}$ is equivalent to $\sqrt[n]{a^m}$. This means instead of using a radical symbol, you can use a fraction as an exponent to express the same idea.
For example, $a^{2/3}$ indicates the cube root of $a^2$. Here:

The numerator (2) tells us the power of "a" inside the radical.
The denominator (3) indicates the type of root (cube root in this case).

With rational exponents, you can convert between radical and exponential forms easily. Converting $\sqrt[3]{a^2}$ to $a^{2/3}$ makes it simpler to apply additional exponent rules, ultimately streamlining computations.

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91影视

Write each expression in exponential form. $$\sqrt[3]{a^{2}}$$

Short Answer

Step by step solution

Converting Radical to Exponential Form

Substitution

Key Concepts

Radicals

Exponents

Rational Exponents

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