Problem 14 - Intermediate Algebra

Question

Fill in the blanks. Write each radical equation using a rational exponent. a. \(\sqrt{x+10}+5=15\) can be written \((x+10)+5=15\) b. \(\sqrt[3]{2 t+4}=t-1\) can be written \((2 t+4)=t-1\)

Step-by-Step Solution

Answer

a. \\((x+10)^{1/2} + 5 = 15\\) b. \\((2t+4)^{1/3} = t - 1\\)

Step 1: Understanding Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. The square root symbol (√) can be written with a rational exponent as a power of 1/2, while a cube root (∛) can be rewritten with a rational exponent of 1/3.

Step 1: Rewrite Square Root as Rational Exponent

In expression a, \(\sqrt{x+10}+5=15\), the square root can be rewritten with a rational exponent. This means that \(\sqrt{x+10}\) is equivalent to \((x+10)^{1/2}\). Thus, we can rewrite the equation as \((x+10)^{1/2} + 5 = 15\).

Step 2: Rewrite Cube Root as Rational Exponent

For expression b, \(\sqrt[3]{2 t+4}=t-1\), the cube root symbol can be expressed with a rational exponent. Hence, \(\sqrt[3]{2t+4}\) is rewritten as \((2t+4)^{1/3}\). Therefore, the expression becomes \((2t+4)^{1/3} = t - 1\).

Key Concepts

Radical Equations

When working with radical equations, you're dealing with equations that contain radicals, such as square roots or cube roots. Understanding how to convert these radicals into expressions with exponents can make manipulating and solving these equations easier.
Radical equations are typically solved by isolating the radical and then eliminating it. Eliminating the radical is usually done by raising both sides of the equation to a power that matches the radical's index.

• If the equation involves a square root, you would square both sides.
• If it involves a cube root, you would cube both sides.

After eliminating the radical, you'll often be left with a polynomial equation. Solving this can involve techniques such as factoring, applying the quadratic formula, or other methods specific to the equation's structure. Keep an eye out for extraneous solutions that can arise when you manipulate radicals.

Square Root

Square roots are one of the most common types of radicals encountered. The square root of a number is a value that, when multiplied by itself, yields the original number.
The notation for the square root of a number, say \(x\), is \(\sqrt{x}\). In terms of exponents, \(\sqrt{x}\) can be rewritten as \(x^{1/2}\). This transformation is particularly useful in algebra when you need to simplify or manipulate expressions.

• For example, solving \(\sqrt{x+10} + 5 = 15\) involves isolating the square root first.
• By moving 5 to the other side, it becomes \(\sqrt{x+10} = 10\).
• Then, square both sides to eliminate the square root, resulting in \(x+10 = 100\).
• Finally, solve for \(x\) by subtracting 10, giving \(x = 90\).

This technique of rewriting the square root as a rational exponent can be helpful in finding the solution quickly.

Cube Root

Cube roots involve finding a number which, when multiplied by itself twice, results in the original number. The cube root symbol is \(\sqrt[3]{x}\), and it can be represented using rational exponents as \(x^{1/3}\).
Using rational exponents helps simplify expressions and equations that involve cube roots. Let's see how this works in an example.

• Consider the equation \(\sqrt[3]{2t+4} = t - 1\).
• By rewriting the cube root using an exponent, it becomes \((2t+4)^{1/3} = t - 1\).
• To solve it, you'll cube both sides: \(2t+4 = (t-1)^3\).
• This gives a polynomial equation, which can then be solved using appropriate algebraic methods.

The ability to express cube roots with rational exponents is powerful and often aids in finding solutions. By understanding how to manipulate these expressions, solving equations with cube roots becomes more straightforward.