Problem 102 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the expression. $$\frac{\left(x^{-2}\right)\left(x^{1 / 2}\right)}{\left(x^{-1}\right)\left(x^{5 / 2}\right)}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{-3}\)

Step 1: Simplify numerator

By using the rule of exponents where \(a^{m} * a^{n} = a^{m+n}\) , the exponent in the numerator is computed. \(x^{-2} * x^{1/2} = x^{-2 + 1/2} = x^{-3/2}\)

Step 2: Simplify denominator

Similarly, simplify the denominator: \(x^{-1} * x^{5/2} = x^{-1 + 5/2} = x^{3/2}\)

Step 3: Divide the terms

Now divide the simplified numerator by the denominator. Use the rule \(a^{m} / a^{n} = a^{m-n}\) : \(x^{-3/2} / x^{3/2} = x^{-3/2 - 3/2} = x^{-3}\)

Key Concepts

Exponents and Powers

Understanding exponents and powers is essential for simplifying algebraic expressions. Exponents, also known as powers, indicate how many times a number, known as the base, is multiplied by itself. For example, in the expression \(x^2\), the base is \(x\), and the power, or exponent, is 2, which means \(x\) is multiplied by itself once: \(x * x\).

When dealing with exponents, there are several types of power expressions to be aware of:

• Positive Integer Exponents: These denote straightforward multiplication, as in \(x^3 = x * x * x\).
• Zero Exponent: Any non-zero base raised to the power of zero is always 1, so \(x^0 = 1\), regardless of the value of \(x\).
• Negative Exponents: These imply division by the base repeated as many times as the value of the negative exponent, for example \(x^{-2}\) means \(1 / (x * x)\).
• Fractional Exponents: A fractional exponent like \(x^{1/2}\) represents the square root of \(x\), and in general, \(x^{m/n}\) represents the n-th root of \(x\) raised to the m-th power.

Through the practice of manipulating expressions with exponents, it becomes easier to simplify algebraic expressions efficiently.

Algebraic Fractions

Algebraic fractions, much like numerical fractions, are ratios of algebraic expressions. The general form of an algebraic fraction is \(\frac{a(x)}{b(x)}\), where \(a(x)\) and \(b(x)\) are algebraic expressions and \(b(x)\) is not equal to zero. Simplifying algebraic fractions usually involves factoring, cancelling common factors, and applying the properties of exponents.

It's important to remember that you can only cancel factors that are common to both the numerator and the denominator and not just common terms. Moreover, when dealing with exponents in algebraic fractions, we must apply the rules of exponents diligently to simplify expressions thoroughly. To master algebraic fractions, one must be comfortable with factoring techniques, such as finding the greatest common factor (GCF) or applying the difference of squares, and have a solid grasp of exponent rules.

Rules of Exponents

The rules of exponents are mathematical guidelines that help simplify expressions with powers. They are critical to understanding how to manipulate and simplify expressions with exponents. A few fundamental rules include:

• Product of Powers: To multiply two powers that have the same base, you add the exponents (\(a^m * a^n = a^{m+n}\)).
• Quotient of Powers: To divide two powers with the same base, you subtract the exponents (\(a^m / a^n = a^{m-n}\)).
• Power of a Power: To raise a power to another power, you multiply the exponents (\(\(a^m\)^n = a^{m*n}\)).
• Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the positive opposite of the exponent (\(a^{-n} = 1/a^n\)).
• Zero Exponent: Any base with an exponent of zero equals one (\(a^0 = 1\)), provided that the base is not zero.

By applying these rules accurately, algebraic expressions with exponents can be simplified, making them easier to work with in various mathematical problems.