Problem 124 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{1}{100^{-5 / 2}} $$

Step-by-Step Solution

Answer

The simplified expression is 100,000.

Step 1: Rewrite the expression using positive exponent

The given expression is \( \frac{1}{100^{-5/2}} \). Whenever you have a negative exponent in the denominator, you can move it to the numerator and change the sign of the exponent. Therefore, the expression becomes \( 100^{5/2} \).

Step 2: Break down the fractional exponent

The exponent \( \frac{5}{2} \) in \( 100^{5/2} \) can be seen as a combination of a square root and a power. Specifically, it can be expressed as \( (100^{1/2})^5 \).

Step 3: Calculate the square root

Calculate the square root of 100, which is \( 100^{1/2} = 10 \).

Step 4: Raise to the power of 5

Take the result from the square root calculation and raise it to the power of 5. So, \( (10)^5 \) is calculated as \( 10 \times 10 \times 10 \times 10 \times 10 = 100000 \).

Key Concepts

Fractional Exponents

Fractional exponents can be a bit daunting at first, but with a clear understanding, they simplify many expressions. A fractional exponent, like \( 100^{5/2} \), represents two main operations that we need to perform: taking a root and an exponentiation. The denominator of the fraction indicates the type of root to take, while the numerator represents the power to which the result should be raised.

• When you see a fractional exponent such as \( a^{m/n} \), you should understand it as \( \sqrt[n]{a^m} \).
• You can also interpret it as \( (a^{1/n})^m \), which means you first take the nth root of \( a \), then raise that result to the mth power.

Mastering this concept allows you to simplify expressions and solve equations more efficiently. It connects directly to understanding roots and powers, which are fundamental in algebra and beyond.

Properties of Exponents

Knowing the properties of exponents is crucial for simplifying expressions and solving mathematical problems. These properties help you manipulate and simplify almost any expression involving exponents. Some fundamental properties include:

• Product of Powers: When multiplying two expressions with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: When dividing two expressions with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{mn} \).
• Negative Exponent: When you have a negative exponent, you take the reciprocal and make the exponent positive: \( a^{-n} = \frac{1}{a^n} \).

These properties can be combined to simplify complex expressions like the one in the original problem. Understanding how to apply these rules can make complex algebraic expressions easier to manage.

Square Roots

The square root is a fundamental mathematical concept, essential for solving various problems in algebra and geometry. Simply put, the square root of a number \( x \) is a value that, when multiplied by itself, gives the number \( x \). For example, the square root of 100 is 10, because \( 10 \times 10 = 100 \).

• The symbol \( \sqrt{} \) represents the square root. The term beneath the square root symbol is called the "radicand."
• Square roots are used to solve quadratic equations, and they often appear in formulas you encounter in mathematics, such as the Pythagorean theorem.
• It is crucial to recognize that the square root function returns the principal (only non-negative) root for positive real numbers. Thus, \( \sqrt{100} \) results in 10, not -10.

Being comfortable with calculating and understanding square roots, especially in conjunction with fractional exponents, can significantly aid in your math prowess.