Problem 65 - Elementary Algebra

Question

Simplify. $$ \sqrt{x^{7}} $$

Step-by-Step Solution

Answer

The answer is \(\text{\root}(x^7)=\text{x}^{7/2}\) with the exponent rewrite simplification.

Step 1: Identify the properties of exponents

Recall that \(\frac{a^m}{a^n}=a^{m-n}\) and \(a^m \times a^n = a^{m+n}\). Exponents can also be handled under square roots. The goal is to simplify \(\frac{\text{exponents}}{\text{square roots}}\).

Step 2: Rewrite the radicand using exponent rules

Represent the square root in terms of exponents: \(\text{\root}(a^m)=a^{m/2}\). In our case, \({a=x}\text{, and} m=7}\. Therefore, \text{\root}({x^7})=x^{7/2}\).

Step 3: Simplify the exponent

The term \(x^{7/2}\) represents the simplified version of \(\text{\root}(x^7)\). Now leave it in the rational form: \text{Answer is} \(x^{7/2}\).

Key Concepts

properties of exponents

Understanding the properties of exponents is crucial for simplifying mathematical expressions. Exponents indicate how many times a number, called the base, is multiplied by itself. Here are some fundamental properties:

• Product of Powers: When multiplying two exponential terms with the same base, you add the exponents, i.e., \(\text{a}^{m} \times \text{a}^{n} = \text{a}^{m+n}\ \).
• Quotient of Powers: When dividing two exponential terms with the same base, you subtract the exponents, i.e., \(\frac{\text{a}^{m}}{\text{a}^{n}} = \text{a}^{m-n}\ \).
• Power of a Power: When raising an exponential term to another power, you multiply the exponents, i.e., \((a^m)^n = a^{mn}\ \).

Mastering these properties makes it easier to handle more complex expressions involving exponents. They are especially useful when paired with roots and radicals.

exponent rules

Exponent rules extend beyond the properties we've discussed, providing the groundwork for more advanced operations.

• Zero Exponent Rule: Any non-zero number raised to the power of 0 is 1, i.e., \(a^{0} = 1\ \).
• Negative Exponent Rule: A negative exponent indicates a reciprocal, i.e., \(a^{-n} = \frac{1}{a^n}\ \).
• Fractional Exponents: A fractional exponent indicates a root, so \(a^{1/n} = \sqrt[n]{a}\ \). For example, \(a^{1/2} = \sqrt{a}\ \).

Using exponent rules makes it straightforward to convert between radical and exponential forms. This conversion is essential for solving exercises like our example, where understanding how to express and manipulate terms is key to simplification.

simplifying square roots

Simplifying square roots involves expressing the root in its simplest terms. The rule \(\sqrt[a]{b^m} = b^{m/a}\ \) helps in this transformation. In our exercise, simplifying \(\sqrt{x^{7}}\ \) means rewriting the square root in exponent form. Applying the rule, we see:

• Express the square root as an exponent: \(\sqrt{x^{7}} = x^{7/2}\ \).
• Simplify the exponent if necessary. In our case, \(x^{7/2}\ \) is already in its simplest form.

Understanding these steps allows you to simplify roots efficiently. This method converts roots to fractional exponents, making it easier to apply the known properties and rules of exponents.