Problem 65 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$x^{3 / 2}=125$$

Step-by-Step Solution

Answer

x = 25

Step 1: Understand the equation

The given equation is \(x^{3/2} = 125\). This means we are solving for a number \(x\) such that when it is raised to the power of \(3/2\), the result is 125.

Step 2: Rewrite the exponent

Rewrite \(x^{3/2}\) in a different form. We can express the power \(3/2\) as \((x^{1/2})^3\), which means \(\sqrt{x}^3\). So, the equation becomes \(\sqrt{x}^3 = 125\).

Step 3: Isolate the square root

To isolate \(\sqrt{x}\), we need to take the cube root of both sides of the equation. Therefore, \(\sqrt{x} = \sqrt[3]{125}\).

Step 4: Evaluate the cube root

Evaluate the cube root of 125. Since \(125 = 5^3\), \(\sqrt[3]{125} = 5\). So, \(\sqrt{x} = 5\).

Step 5: Solve for x

To find \(x\), square both sides of the equation \(\sqrt{x} = 5\). Therefore, \(x = 5^2 = 25\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. The exponent tells us how many times to multiply the base by itself. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent. This means multiplying 2 by itself 3 times: \(2 \times 2 \times 2 = 8\).

If the exponent is fractional, it can represent roots as well. For instance, \(x^{3/2}\) involves both a power and a root. We can rewrite \(x^{3/2}\) as \((x^{1/2})^3\) which means the square root of \(x\) raised to the third power. Exponents help simplify expressions and solve equations that would be complex to handle otherwise.

Radicals

Radicals involve roots and represent operations that are the inverse of exponentiation. The most common radicals are the square root and cube root.

The symbol for a square root is \(\sqrt{}\), and it means finding a number that, when multiplied by itself, equals the original number. For instance, \(\sqrt{25} = 5\) because \(5 \times 5 = 25\). The cube root is represented as \(\sqrt[3]{}\) and involves finding a number that, when multiplied by itself three times, equals the original number. For example,\(\sqrt[3]{27} = 3\) because \(3 \times 3 \times 3 = 27\). Understanding radicals is crucial in solving equations where the unknown is under a root sign.

Cube Root

A cube root is found by identifying a number which, when multiplied by itself three times, gives the original number. The cube root of \(x\) can be written as \(\sqrt[3]{x}\), and it is the opposite of raising a number to the power of 3.

For example, to find \(\sqrt[3]{125}\), we need a number that multiplies by itself three times to give 125. We know that \(125 = 5 \times 5 \times 5\), so \(\sqrt[3]{125} = 5\). Cube roots are useful when solving equations where variables are raised to the third power or require simplification involving cubes.

Square Root

The square root of a number is a value that, when multiplied by itself, yields the original number. It is represented by the symbol \(\sqrt{}\). For example, \(\sqrt{16} = 4\) because \(4 \times 4 = 16\).

Square roots are essential in solving equations with powers of 2. When you isolate \(\sqrt{x}\), like in the equation \(\sqrt{x} = 5\), you can find \(x\) by squaring both sides. So, \(\sqrt{x} = 5\) implies \(x = 5^2 = 25\). Understanding square roots allows you to reverse the effect of squaring a number, making them crucial for solving quadratic equations and other algebraic problems.