Problem 593 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(b^{10}\right)^{\frac{3}{5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( b^6 \).

Step 1: Understand the Power Rule

To simplify the expression \( \left(b^{10}\right)^{\frac{3}{5}} \), use the power rule which states that \( \left(a^m\right)^n = a^{m \cdot n} \).

Step 2: Apply the Power Rule

Apply the power rule to the given expression \( \left(b^{10}\right)^{\frac{3}{5}} \). This becomes \( b^{10 \cdot \frac{3}{5}} \).

Step 3: Simplify the Exponent

Multiply the exponents 10 and \( \frac{3}{5} \): \[ 10 \cdot \frac{3}{5} = \frac{30}{5} = 6 \].

Step 4: Write the Final Expression

After simplifying the exponent, the expression becomes \( b^6 \).

Key Concepts

Power Rule

The Power Rule is essential in algebra when dealing with exponents. It states: \( \left(a^m\right)^n = a^{m \cdot n} \). This means if you have an exponent raised to another exponent, you multiply the two exponents together.
For instance, in the exercise where we need to simplify \( \left(b^{10}\right)^{\frac{3}{5}} \), we apply the Power Rule. This results in \( b^{10 \cdot \frac{3}{5}} \).
By understanding this rule, you can easily manage complex expressions involving exponents.

Exponents

Exponents are a way to express repeated multiplication of the same number. For example, \( b^{10} \) means multiplying \( b \) by itself 10 times.
When working with exponents, some key rules to remember are:

• The Power Rule: \( \left(a^m\right)^n = a^{m \cdot n} \)
• Product Rule: \( a^m \cdot a^n = a^{m+n} \)
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)

Applying these rules will help simplify even the most complicated expressions involving exponents.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In algebra, expressions like \( b^{10} \) are common.
Simplifying algebraic expressions often involves applying various rules and properties. For example, the power rule is frequently used to simplify expressions with exponents.
By understanding and applying these rules, students can solve algebraic problems more efficiently. This includes handling expressions like \( \left(b^{10}\right)^{\frac{3}{5}} \) and simplifying them to \( b^6 \).