Problem 21 - Algebra 2

Question

Write each expression in exponential form. $$ (\sqrt{7 x})^{3} $$

Step-by-Step Solution

Answer

The expression \((\sqrt{7 x})^{3}\) in exponential form is \((7x)^{3/2}\).

Step 1: Rewrite the square root in exponential form

This involves expressing the square root in exponential form. A square root is a half power, so \(\sqrt{7x}\) can be written as \((7x)^{1/2}\). The expression then becomes \(((7x)^{1/2})^{3}\).

Step 2: Multiply the exponents

Next, apply the power of a power rule in exponents which states that \((a^{m})^{n} = a^{m*n}\) to simplify the expression. Multiply \(1/2\) by \(3\) to simplify the exponential formula. The expression becomes \((7x)^{3/2}\).

Step 3: Write the final expression

The final expression in exponential form is \((7x)^{3/2}\). This is the most simplified form with the square root converted to fractional exponent, raised to the 3rd power.

Key Concepts

Understanding Square Roots

Square roots can sometimes seem a bit tricky, but they're easier to grasp when you think of them as exponents. Essentially, a square root is a number that, when multiplied by itself, gives the original number under the square root symbol. By definition, the square root of a number is the same as raising it to the power of one-half. For example, the square root of any value, say \(a\), can be expressed as \(a^{1/2}\). This conversion is extremely useful when dealing with expressions that combine square roots and exponents, as it allows for easier simplification and computation.

• This understanding simplifies operations like multiplication and division involving square roots.
• It also allows for the use of exponent rules to further simplify expressions.

By representing square roots as fractional exponents, we can seamlessly integrate them into exponential expressions and make use of the power rules for exponents.

The Basics of Exponents

Exponents are a shorthand way of expressing repeated multiplication. When a number or variable is multiplied by itself, we use an exponent to indicate how many times it is used as a factor. For instance, \(x^3\) means that \(x\) is multiplied by itself three times, or \(x \times x \times x\). Exponents allow us to work efficiently with larger numbers and complex expressions.

• They reflect how many times a base number is used in a multiplication.
• Using exponents, we can perform quick arithmetic operations.

Understanding how to handle exponents is vital, especially when they represent variables or are part of larger equations. They provide a powerful tool for algebraic manipulation and simplification, especially when applying the rules of exponents like multiplication, division, and power of a power.

Applying the Power of a Power Rule

The power of a power rule is a fundamental property in algebra that simplifies expressions with exponents. If you have a base number raised to an exponent, and then the entire expression raised again to another exponent, you can simplify it using this rule. Mathematically, this rule is expressed as \((a^m)^n = a^{m \times n}\). It tells us that when you take an exponent to another exponent, you simply multiply the exponents.

• This rule helps in reducing complex exponential expressions.
• It's incredibly handy for simplifying nested exponents.

For example, in the expression \(((7x)^{1/2})^3\), you apply the power of a power rule by multiplying \(1/2\) and \(3\), resulting in \(7x^{3/2}\). This powerful rule ensures that you can break down and simplify expressions efficiently, making calculations more straightforward.