Problem 51 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{b^{1 / 2} b^{3 / 4}}{-b^{1 / 4}} $$

Step-by-Step Solution

Answer

-b

Step 1: Multiply the Exponents in the Numerator

To simplify the expression in the numerator, apply the property of exponents that states \(a^m \cdot a^n = a^{m+n}\). Here, we have \(b^{1/2} \cdot b^{3/4}\). This turns into:\[b^{1/2 + 3/4} = b^{2/4 + 3/4} = b^{5/4}.\]

Step 2: Divide the Simplified Numerator by the Denominator

Now, divide \(b^{5/4}\) (from Step 1) by the denominator \(-b^{1/4}\). Use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\). This becomes:\[\frac{b^{5/4}}{-b^{1/4}} = -b^{5/4 - 1/4} = -b^{4/4} = -b^1.\]

Step 3: Simplify the Final Expression

Since \(-b^1\) is equal to \(-b\), there is no further simplification needed. The final simplified expression is \(-b\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that helps reduce complex expressions to their simplest form. This action makes it easier to work with the expression in further calculations. To simplify an expression involving exponents, it's important to use the properties of exponents methodically. There are certain rules we can apply. For instance:

• The product of powers rule: when multiplying two powers with the same base, you add their exponents. As seen in our example, we have \(b^{1/2} \cdot b^{3/4} = b^{1/2+3/4}\) or \(b^{5/4}\).
• The quotient of powers rule: when dividing two powers with the same base, you subtract the exponents. In the example, \(\frac{b^{5/4}}{b^{1/4}} = b^{5/4-1/4}\) simplifies to \(b^{4/4} = b^1\).

Simplifying expressions correctly by applying these rules can make solving algebraic problems more straightforward.

Positive Exponents

Positive exponents indicate the number of times a base is multiplied by itself. In mathematics, it is ideal practice to write expressions with positive exponents. This ensures clarity and standardization in solving equations across various contexts.
When an exponent is positive, it essentially represents repeated multiplication. In our problem, we are asked to express the final result with positive exponents. After simplifying the given expression using exponent rules, the final term was \(-b^1\). The exponent \(1\) is already positive, fulfilling the requirement.
Writing expressions with positive exponents can help avoid confusion and reduce errors, especially in more complex algebraic manipulations.

Rational Exponents

Rational exponents, also known as fractional exponents, describe both the power and root of a number. Simply put, if you have an expression \(a^{m/n}\), it equates to the \(n\)-th root of \(a\) raised to the power \(m\).
Understanding rational exponents involves knowing their dual role: \(b^{1/2}\) implies "the square root of \(b\)", and \(b^{3/4}\) means "the fourth root of \(b^3\)". This can sometimes lead to more complex initial expressions, like in our example.
By treating these fractional exponents just like integer exponents when using properties like multiplication and division, we can simplify expressions systematically. In our task, converting the expression to positive rational exponents through these properties was key to reaching the simplified form. Thus, grasping how to manage rational exponents is fundamental to working efficiently with algebraic expressions.