Problem 76 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$3 x^{3 / 4}=x^{1 / 2}$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{81} \).

Step 1: Rewrite the equation

Rewrite the given equation to isolate the variable terms on both sides. The equation is: \[ 3 x^{3 / 4} = x^{1 / 2} \]

Step 2: Divide both sides by \( x^{1 / 2} \)

To isolate the powers of \( x \), divide both sides of the equation by \( x^{1 / 2} \): \[ \frac{3 x^{3 / 4}}{x^{1 / 2}} = \frac{x^{1 / 2}}{x^{1 / 2}} \] Which simplifies to: \[ 3 x^{(3 / 4 - 1 / 2)} = 1 \]

Step 3: Simplify the exponent

Subtract the exponent in the denominator from the exponent in the numerator: \[ 3 x^{(3 / 4 - 2 / 4)} = 1 \] Simplify the exponent: \[ 3 x^{1 / 4} = 1 \]

Step 4: Isolate \( x^{1 / 4} \)

Divide both sides by 3 to solve for \( x^{1 / 4} \): \[ x^{1 / 4} = \frac{1}{3} \]

Step 5: Solve for \( x \)

Raise both sides to the power of 4 to eliminate the fraction exponent: \[ x = \frac{1}{3^4} \] Which simplifies to: \[ x = \frac{1}{81} \]

Key Concepts

Rational Exponents

Rational exponents are exponents that are fractions. They represent both roots and powers in a more compact form. For instance, the expression \( x^{1/2} \) is the same as \( \sqrt{x} \), which means the square root of \( x \). Similarly, \( x^{3/4} \) represents the fourth root of \( x^3 \).

When dealing with rational exponents, it is essential to understand how to manipulate them. For example:

• \( a^{m/n} = \sqrt[n]{a^m} \)
• \( (a^m)^{1/n} = \sqrt[n]{a^m} = a^{m/n} \)
• \( a^{m/n} \cdot b^{m/n} = (a \cdot b)^{m/n} \)

Rational exponents obey all the usual rules of exponents, making them powerful tools in equation solving and simplification.

Equation Simplification

Equation simplification involves breaking down complex expressions into simpler ones. This process often makes it easier to solve equations. It includes combining like terms, factoring, and reducing fractions. In the given example:

The original equation is \( 3 x^{3 / 4} = x^{1 / 2} \).

We start by dividing both sides by \( x^{1 / 2} \). This is a crucial step because it allows us to consolidate the exponents. After division, the equation becomes:

• \[ \frac{3 x^{3 / 4}}{x^{1 / 2}} = 1 \]

Using properties of exponents, subtracting \( 1 / 2 \) from \( 3 / 4 \) results in \( 1 / 4 \) because \(\frac{3}{4}\) euros is equivalent to 0.75, and \(\frac{1}{2}\) euros is equivalent to 0.5, thus:

• \[ 3 x^{1/4} = 1 \]

Then, by isolating \( x \), we first divide by 3:

• \[ x^{1/4} = \frac{1}{3} \]

Finally, raising both sides to the power of 4 eliminates the exponent:

• \[ x = \frac{1}{3^4} = \frac{1}{81} \]

Fraction Exponents

Fraction exponents (or rational exponents) provide a way to write roots and powers in one compact form. For instance, the expression \( x^{1/n} \) represents the nth root of \( x \), such as \( x^{1/2} \) for the square root.

Here are some important properties of fraction exponents:

• \( x^{a/b} \) is equivalent to \( (x^a)^{1/b} \) or \( \sqrt[b]{x^a} \).
• \( x^{a/b} \times x^{c/d} = x^{(ad+bc)/bd} \)
• \( (x^{a/b})^c = x^{ac/b} \)

In our given solution, when simplifying \( 3 x^{3/4} = x^{1/2} \), we divide by \( x^{1/2} \), which involves understanding that you can subtract the fractions in the exponent:

• \( 3 x^{(3/4 - 1/2)} = 1 \)

Subtraction of the fractions (\( 3/4 - 1/2 = 3/4 - 2/4 = 1/4 \)) makes this simplification manageable, leading us to the simpler form of the expression, \( x^{1/4} \).