Problem 20 - Algebra 2

Question

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

Step-by-Step Solution

Answer

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

Step 1: Examine the expression

Here we have the expression \(\sqrt{{(7x)^3}}\) to work with. It is a square root of a term raised to a power.

Step 2: Apply radical and exponent rules

We know that the square root function is equivalent to raising to the power of \(\frac{1}{2}\). Also, when there's a power to a power, we multiply them. Here \(7x\) is raised to the power 3 and this entire expression is under a square root, meaning it's raised to the power \(\frac{1}{2}\). Therefore, we can rewrite the expression as \((7x)^{3 \cdot \frac{1}{2}}\).

Step 3: Simplify the exponent

Multiplying the exponents, we get \((7x)^{\frac{3}{2}}\).

Key Concepts

Understanding Radicals

Radicals are mathematical symbols used to indicate roots of numbers. The most common is the square root, represented with the radical sign \(\sqrt{\cdot}\). Radicals are involved when we're looking to find a value that, when multiplied by itself a certain number of times, gives the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\).
When dealing with radicals, it is important to recognize that they can also be expressed as exponents. Specifically, a square root can be rewritten as raising a number to the power of \(\frac{1}{2}\). This is useful for simplifying expressions involving roots, as it allows us to leverage exponential rules. For example, \(\sqrt{(7x)^3}\) can be converted to \(((7x)^3)^{\frac{1}{2}}\). This step is crucial for simplifying complex expressions using exponent rules.

Exponents and Their Rules

Exponents are used to denote repeated multiplication of a number by itself. An expression like \(a^n\) signifies that \(a\) is multiplied by itself \(n\) times. This concise notation helps in simplifying expressions and solving equations, especially when dealing with powers of a number.
There are several rules for working with exponents that are key when simplifying expressions:

• Product of Powers Rule: \(a^m \times a^n = a^{m+n}\)
• Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
• Power of a Product Rule: \((ab)^n = a^n \times b^n\)

Using these rules can transform complex expressions into simpler forms. In our exercise, we utilized the "Power of a Power" rule to deal with \(\sqrt{(7x)^3}\), converting it into \((7x)^{3 \cdot \frac{1}{2}}\), and eventually simplifying it to \((7x)^{\frac{3}{2}}\). Understanding these rules is crucial for both solving and simplifying mathematical expressions effectively.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing an equation or expression to its simplest form. This means making an expression easier to work with and understand, often by eliminating radicals or combining like terms.
In the context of exponents and radicals, simplifying involves applying the exponent rules effectively. As we saw in the exercise, expressing radicals as exponents allowed us to use the power rules, particularly the "Power of a Power Rule", which tells us to multiply exponents together. This helps in rewriting the expression \((7x)^{\frac{3}{2}}\) as its simplest form, showcasing the power that rules of exponents can provide in simplification.
Mastering the process of simplifying expressions not only makes solving algebraic equations easier but also aids in revealing the properties and relationships inherent in mathematical expressions.