Problem 19 - Algebra 2 and Trigonometry

Question

In \(18-23,\) solve for the variable in each equation. Express the solution to the nearest hundredth. $$ y^{\frac{2}{3}}=6 $$

Step-by-Step Solution

Answer

The solution to the equation is approximately 14.70.

Step 1: Isolate the term with the variable

The given equation is \( y^{\frac{2}{3}}=6 \). To solve for \( y \), we first need to undo the exponent with a reciprocal exponent.

Step 2: Apply the cube exponent

To eliminate the fractional exponent, raise both sides of the equation to the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This gives us \( (y^{\frac{2}{3}})^{\frac{3}{2}}=(6)^{\frac{3}{2}} \).

Step 3: Simplify the left side

Simplifying the left side, we get \( y^{\frac{2}{3} \times \frac{3}{2}} = y^1 = y \).

Step 4: Simplify the right side

On the right side, \( (6)^{\frac{3}{2}} \) can be broken down as \( (\sqrt{6})^3 \). First, calculate \( \sqrt{6} \), which is approximately 2.45. Then cube this to find \( 2.45^3 \).

Step 5: Calculate \( 2.45^3 \) and round to the nearest hundredth

Compute \( 2.45^3 \), which is approximately 14.70. Therefore, \( y \approx 14.70 \).

Key Concepts

Rational Exponents

Rational exponents are a different way of expressing powers and roots in mathematics. They serve as a bridge between integer exponents and radicals.
Understanding rational exponents is key to solving many algebraic problems.

• **Rational exponents** are expressed as fractions. For example, in the expression \( y^{\frac{2}{3}} \), **2** is the numerator, and **3** is the denominator.
• The **numerator** tells us the power we need to raise the base number to. So, in \( y^{\frac{2}{3}} \), the base \( y \) is raised to the power of 2.
• The **denominator** represents the root that must be extracted from the base number. For \( y^{\frac{2}{3}} \), this means taking the cube root of \( y^2 \).

To reverse a rational exponent like \( \frac{2}{3} \), apply its **reciprocal**, \( \frac{3}{2} \). By raising both sides of an equation with rational exponents to the reciprocal, you can effectively eliminate the fractional power and simplify the expression.

Exponentiation

Exponentiation is a fundamental operation in mathematics that involves raising a number to a power. It's a shortcut for repeated multiplication.
Understanding how to manipulate exponents is crucial for solving exponential equations.
**Handling Exponential Equations**

• When you **raise a power to another power**, you multiply the exponents. This rule helps simplify complex expressions: \( (a^m)^n = a^{m \times n} \).
• **Finding reciprocal exponents** is useful when dealing with fractional powers. For instance, to get rid of an exponent \( \frac{2}{3} \), you use \( (x^{\frac{2}{3}})^{\frac{3}{2}} = x \).
• To solve equations like \( (6)^{\frac{3}{2}} \), break it down: take the square root first, \( \sqrt{6} \), and then raise it to the power of 3.

By mastering these concepts, you can effectively solve exponential equations and simplify expressions with ease.

Simplifying Expressions

Simplifying expressions is an essential skill when solving equations, particularly those involving exponents.
It involves rewriting expressions in a simpler, more concise form.
**Steps for Simplification**

• Begin by **isolating** the term with a variable or power, if possible.
• Simplify one side at a time, whether it's breaking down radicals or multiplying out powers.
• For an expression like \( (\sqrt{6})^3 \), first compute the square root, then apply the exponent by multiplying: \( 2.45 \times 2.45 \times 2.45 \).
• Always round final answers to the required level of precision, such as to the nearest hundredth, ensuring your final answer is both accurate and precise.

Through consistent practice, simplifying expressions will become a straightforward process, allowing for quicker and more accurate solutions across various mathematical problems.