Problem 39 - Algebra and Trigonometry

Question

Simplify the expression. Assume the letters denote any real numbers. \(\sqrt[5]{a^{6} b^{7}}\)

Step-by-Step Solution

Answer

The simplified expression is \(a^{\frac{6}{5}} b^{\frac{7}{5}}\).

Step 1: Apply the Power of a Power Rule

The given expression is \(\sqrt[5]{a^{6} b^{7}}\). We can start by using the property of exponents known as the "power of a power" rule. This rule states that \((x^m)^n = x^{m \cdot n}\). Applying this to the roots, we can rewrite \(\sqrt[5]{x}\) as \(x^{\frac{1}{5}}\). So, we rewrite the expression as \((a^6 b^7)^{\frac{1}{5}}\).

Step 2: Distribute the Exponents

Using the rule \((xy)^n = x^n y^n\), distribute the exponent \(\frac{1}{5}\) to each term inside the parentheses. This will give us \(a^{6 \times \frac{1}{5}} b^{7 \times \frac{1}{5}}\).

Step 3: Simplify the Exponents

Calculate the new exponents by multiplying the exponents: \(a^{6 \times \frac{1}{5}} = a^{\frac{6}{5}}\) and \(b^{7 \times \frac{1}{5}} = b^{\frac{7}{5}}\).

Step 4: Write the Simplified Expression

Combine the simplified exponents into a single expression: \(a^{\frac{6}{5}} b^{\frac{7}{5}}\). This represents the simplified form of the original expression.

Key Concepts

Power of a power rule

The power of a power rule is an essential exponent rule for simplifying expressions involving exponents. This rule states that when you raise a power to another power, you multiply the exponents together. In mathematical terms, this is expressed as \[ (x^m)^n = x^{m \cdot n}. \] This concept is particularly useful when dealing with roots and exponents because it allows you to transform complex expressions into simpler forms by managing the exponents directly.
Let's apply this to our example: if you need to simplify \( \sqrt[5]{x} \), it can also be written as \( x^{\frac{1}{5}} \). Thus, the expression \( \sqrt[5]{a^6 b^7} \) becomes \( (a^6 b^7)^{\frac{1}{5}} \). Using the power of a power rule simplifies the expression by consolidating exponents, making it easier to manipulate and eventually simplify further.

Exponent rules

Exponent rules are foundational guidelines that help in manipulating expressions involving powers. Understanding these rules allows you to simplify expressions and solve equations efficiently.
Key exponent rules include:

• Product of Powers Rule: When you multiply powers with the same base, add their exponents. \( x^m \cdot x^n = x^{m+n} \).
• Quotient of Powers Rule: When you divide powers with the same base, subtract their exponents. \( \frac{x^m}{x^n} = x^{m-n} \).
• Power of a Product Rule: When raising a product to a power, apply the exponent to each factor in the product separately. \( (xy)^n = x^n y^n \).

In our example, we use the power of a product rule to distribute the fractional exponent \( \frac{1}{5} \) to each term in the expression \( (a^6 b^7)^{\frac{1}{5}} \). This results in each term receiving the exponent: \( a^{6 \times \frac{1}{5}} b^{7 \times \frac{1}{5}} = a^{\frac{6}{5}} b^{\frac{7}{5}} \). By applying the correct exponent rules, you simplify the expression to its core components.

Radical expressions

Radical expressions involve roots, such as square roots, cube roots, or in a more general form, any \( n \)-th root \( \sqrt[n]{x} \). Simplifying radical expressions often involves converting them into expressions with fractional exponents for easier manipulation.
The conversion uses the understanding that a radical expression \( \sqrt[n]{x^m} \) can be rewritten using exponents as \( x^{\frac{m}{n}} \). This makes the computation more straightforward since you can now apply exponent rules to simplify the expression further.
In our example, \( \sqrt[5]{a^6 b^7} \) translates to \( (a^6 b^7)^{\frac{1}{5}} \), and further simplifying gives \( a^{\frac{6}{5}} b^{\frac{7}{5}} \). Through these steps, the expression becomes clearer and more usable in subsequent mathematical operations or real-life applications.