Problem 64 - Applied Calculus

Question

Write each expression in power form \(a x^{b}\) for numbers \(a\) and \(b\). $$ \sqrt[3]{\frac{8}{x^{6}}} $$

Step-by-Step Solution

Answer

The expression in power form is \(2x^{-2}\).

Step 1: Convert the Expression into Power Form

First, express the cube root as an exponent. The cube root of any number can be written as raising that number to the power of \(\frac{1}{3}\). Thus, the original expression \(\sqrt[3]{\frac{8}{x^{6}}}\) becomes \((\frac{8}{x^6})^{1/3}\).

Step 2: Apply Power to Each Factor

Use the property of exponents which states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\). Apply the \(\frac{1}{3}\) exponent to both the numerator and the denominator of \(\frac{8}{x^6}\). Thus, \((8)^{1/3}\) and \((x^6)^{1/3}\).

Step 3: Simplify the Numerator

Calculate \((8)^{1/3}\). Since \(8 = 2^3\), we have \((2^3)^{1/3} = 2^{(3 \cdot \frac{1}{3})} = 2^1 = 2\). Thus, the numerator simplifies to 2.

Step 4: Simplify the Denominator Exponent

Calculate \((x^6)^{1/3}\). Using the exponent rule \((x^m)^n = x^{mn}\), we get \((x^6)^{1/3} = x^{6 \cdot \frac{1}{3}} = x^2\). Thus, the denominator simplifies to \(x^2\).

Step 5: Combine Simplified Parts

Combine the simplified numerator and denominator. Thus, the fraction \(\frac{2}{x^2}\) can be written in the form of \(2 \cdot x^{-2}\) using negative exponents, or as \(2x^{-2}\) using power form with \(a = 2\) and \(b = -2\).

Key Concepts

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. When you see an expression like \(x^n\), it means "multiply \(x\) by itself \(n\) times." For example, \(3^2\) is \(3 \/ \times \/ 3 = 9\). Exponents have some fundamental rules that make it easier to work with powers in math:

• Product of Powers Rule: When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of Powers Rule: When dividing like bases, subtract the exponents: \(a^m / a^n = a^{m-n}\)
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\)

Exponents can also be fractions, which allow us to express roots, such as square roots or cube roots, in power form.

Cube Root

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. It's another way to "undo" the operation of cubing a number. In power form, a cube root is expressed as raising a number to the power of \(\frac{1}{3}\). For example, the cube root of 8, written \(\sqrt[3]{8}\), is 2 because \(2^3 = 8\).
In mathematical operations, using cube roots might be necessary to simplify expressions or solve equations. Transitioning between radical notation and exponential notation allows us to apply exponent rules, easing calculations. For example, expressing \(\sqrt[3]{x}\) as \(x^{1/3}\) allows us to easily use exponent laws for further calculations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This can make it easier to work with them or understand their behavior. Key strategies include:

• Combining Like Terms: Terms that have the same variable and exponent can be combined by adding or subtracting their coefficients.
• Using Exponent Rules: Apply rules like the Product of Powers and Power of a Power to condense expressions.
• Factoring: Breaking down complex expressions into products of simpler ones can facilitate simplification.

When dealing with expressions such as \(\sqrt[3]{\frac{8}{x^6}}\), simplifying involves rewriting parts of the expression using exponent laws, calculating powers or roots, and combining results. This ensures that the expression is in a manageable form like \(2x^{-2}\).

Negative Exponents

Negative exponents represent the concept of reciprocal. A negative exponent indicates that the base should be moved to the opposite part of a fraction, essentially flipping it. For example, \(a^{-n} = \frac{1}{a^n}\). It's a way to express division in terms of multiplication.
This concept helps in simplifying expressions in algebra, especially when working with fractions or writing expressions in power form. For instance, the expression \(2x^{-2}\) implies that \(x^2\) is in the denominator, translated back into a fraction as \(\frac{2}{x^2}\).
Using negative exponents simplifies the process of managing complex expression and gives us another tool to rearrange terms efficiently.