Problem 90 - College Algebra

Question

In Exercises \(85-94,\) simplify using properties of exponents. $$\left(x^{4 / 5}\right)^{5}$$

Step-by-Step Solution

Answer

The simplified version of the given expression \((x^{4 / 5})^5\) is \(x^{4}\).

Step 1: Recognize the Exponential Rule

The given expression is a power raised to another power. The general rule for such exponential expressions is \((a^{m})^{n} = a^{m*n}\).

Step 2: Apply the Exponential Rule

Now we apply the rule to the given expression. We have \(x^{(4 / 5)*5}\). We can simplify by multiplying the exponents \(4 / 5\) and \(5\).

Step 3: Simplify the Expression

Multiplying \(4 / 5\) by \(5\), we get \(4\). Hence, \((x^{4 / 5})^5 = x^{4}\).

Key Concepts

Exponent Rules

Exponents allow us to express repeated multiplication in a concise form. Understanding exponent rules is crucial for simplifying expressions. One key rule is the **product of powers rule**, which states when you multiply same bases, you can add the exponents:

• \[a^m imes a^n = a^{m+n}\]

The **quotient of powers rule** tells us we can subtract exponents if the bases are the same:

• \[\frac{a^m}{a^n} = a^{m-n}\]

Another vital rule is the **power of a power rule**, which is used when an exponent is raised to another power. This rule is particularly useful in our original exercise:

• \[(a^{m})^{n} = a^{m\times n}\]

These rules simplify the process of working with exponents and are essential for tackling more advanced algebra problems.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process often makes them easier to understand and solve. For exponent expressions, this means applying the relevant exponent rules to combine or reduce terms. In expressions involving powers, always look for opportunities to use multiplication or division rules. This allows you to transform an expression such as

• \[(x^{4/5})^5\]

into a simpler form like

• \[x^4\]

Breaking down expressions step-by-step while keeping these rules in mind not only simplifies the calculations but also aids in understanding each component of the expression. It’s important to practice this regularly, as this lays a foundational understanding in algebra and beyond.

Multiplying Exponents

Multiplying exponents is straightforward but needs to be done carefully. When multiplying terms with the same base but different exponents, you simply add the exponents. For example:

• \[a^m imes a^n = a^{m+n}\]

However, when dealing with a scenario like in our original problem where a term like

• \[a^{\frac{4}{5}} \times a^{5}\]

would be handled by first converting it using the power of a power rule. Always ensure you multiply exponents directly rather than adding, in the case of powers being raised to powers. This careful multiplication, as seen in the simplification process, helps to avoid errors and keeps your math neat and orderly.

Power of a Power Rule

The power of a power rule is an important exponent rule to remember. It describes an expression where a power is raised to another power. For instance, in the expression

• \[(x^{4/5})^5\]

we apply this rule. According to the rule,

• \[(a^{m})^{n} = a^{m \times n}\]

This involves multiplying the outer exponent by the inner exponent, leading to simpler expressions. In our exercise example, it reduces the expression to

• \[x^4\]

This method is particularly useful when working with algebraic expressions and helps consolidate the problem into a manageable solution. Always be sure to apply the power of a power rule whenever you encounter an exponent raised to another exponent, simplifying your work efficiently.