Problem 46 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ \sqrt[3]{y^{5}} $$

Step-by-Step Solution

Answer

\( y^{\frac{5}{3}} \)

Step 1: Understand the Expression

We need to simplify the expression \( \sqrt[3]{y^5} \). This means we are looking for the simplest form of \( y^5 \) raised to the power of \( \frac{1}{3} \), since taking a cube root is the same as raising to the one-third power.

Step 2: Apply the Property of Exponents

We use the property \( (a^m)^n = a^{m \cdot n} \). Here, \( m = 5 \) and \( n = \frac{1}{3} \), so we substitute into the formula which gives us \( y^{5 \cdot \frac{1}{3}} \).

Step 3: Multiply the Exponents

Multiply the exponents: \( 5 \cdot \frac{1}{3} \) to obtain \( \frac{5}{3} \). This means the expression simplifies to \( y^{\frac{5}{3}} \).

Step 4: Simplify Further If Needed

The expression \( y^{\frac{5}{3}} \) is fully simplified. It can also be expressed as \( y^{1\frac{2}{3}} = y^{1} \cdot y^{\frac{2}{3}} \), which shows both the whole number part and the fractional exponent part.

Key Concepts

Understanding Cube Roots

A cube root is a mathematical operation that finds a number that, when multiplied by itself three times, gives the original number. It is the inverse operation of cubing a number. The symbol for the cube root is \( \sqrt[3]{\ } \). For example, the cube root of 8 is 2, because 2 multiplied by itself three times results in 8.
\[ 2 \times 2 \times 2 = 8 \]
In algebra, cube roots can be used to simplify expressions involving variables raised to powers. Consider an expression like \( \sqrt[3]{y^5} \). To find the cube root, we think of the exponent of \( y \) as being divided by 3. This is equivalent to raising \( y^5 \) to the power of \( \frac{1}{3} \). This brings us to the idea of fractional exponents, a helpful concept in dealing with roots.

Properties of Exponents

Exponents are powerful notations in mathematics that indicate how many times a number, the base, is multiplied by itself. Understanding the properties of exponents helps in simplifying complex expressions. Some fundamental properties include:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power: \((a^m)^n = a^{m \cdot n} \)

When dealing with expressions like \( (y^5)^{\frac{1}{3}} \), the property "power of a power" is especially useful. It helps us rearrange and simplify expressions by multiplying the exponents. For \((y^5)^{\frac{1}{3}}\), we multiply the 5 by \( \frac{1}{3} \) to get \( y^{\frac{5}{3}} \). This makes the expression much easier to manage and understand.

Diving Into Fractional Exponents

Fractional exponents are another way to represent roots. They combine the notions of powers and roots into a single expression. The general form is \( a^{\frac{m}{n}} \), which is equivalent to the \( n\)-th root of \( a^m \), or \( \sqrt[n]{a^m} \). For example, \( 8^{\frac{2}{3}} \) means the cube root of \( 8^2 \), or \( \sqrt[3]{64} \), which simplifies down to 4.
In our specific problem, we simplify \( y^{\frac{5}{3}} \). Here, 5 is your numerator, indicating the power. The denominator 3 indicates the cube root. Therefore, \( y^{\frac{5}{3}} \) can be broken down as \( y^1 \times y^{\frac{2}{3}} \). Understanding fractional exponents is crucial as they provide a compact way to express both powers and roots, making calculations more concise.