Problem 4 - Elementary and Intermediate Algebra

Question

Concept Reinforcement In each of Exercises \(1-8\), match the expression with an equivalent expression from the column on the right. Assume \(a, b>0\) a) \(\frac{\sqrt[5]{a^{2}} \sqrt[5]{b^{2}}}{\sqrt[5]{b^{5}}}\) b \(\frac{a^{2}}{b^{3}}\) c) \(\sqrt{\frac{a \cdot b}{b^{3} \cdot b}}\) d) \(\sqrt{a}\) e) \(\frac{\sqrt[3]{a^{2}}}{b^{2}}\) f) \(\sqrt[5]{\frac{a^{6} b}{b^{4} \cdot b}}\) g) \(2 a\) h) \(\frac{\sqrt[5]{a^{2} b^{3}}}{\sqrt[5]{b^{5}}}\) $$ -\sqrt{\frac{a}{b^{3}}} $$

Step-by-Step Solution

Answer

a: \(\frac{\sqrt[5]{a^{2}b^{2}}}{b}\), b: \(\frac{a^{2}}{b^{3}}\), c: \(\frac{a}{b^{3}}\), d: \(\frac{\sqrt{a}}{\underline{\phantom{xx}}}{b^{3}}\), e: \(\frac{\sqrt[3]{a^{2}}}{b^{2}}\), f: \(\frac{\sqrt{a^{6}b}{b^{4}}}{b^{4}}\), g: \frac{a}{b^{2}}, h: \frac{a^{2}[5]^{[a^{2}b^{3]}^{a^{5}}].[b^{5}}}\frac[5][a^{b}}.fraction[a}{[\frac{a^{2}{b}{b^{2}}}}

Step 1: Simplify expression a

Expression a is \(\frac{\sqrt[5]{a^{2}} \sqrt[5]{b^{2}}}{\sqrt[5]{b^{5}}}\). Combine the fifth roots in the numerator: \(\frac{\sqrt[5]{a^{2}b^{2}}}{\sqrt[5]{b^{5}}}\). Simplify inside the fifth root: \(\frac{\sqrt[5]{a^{2}b^{2}}}{b}\).

Step 2: Simplify expression b

Expression b is \(\frac{a^{2}}{b^{3}}\). This is already simplified and cannot be reduced further.

Step 3: Simplify expression c

Expression c is \(\sqrt{\frac{a \cdot b}{b^{3} \cdot b}}\). Simplify inside the square root: \(\sqrt{\frac{a \cdot b}{b^{4}}}\). This simplifies to \(\sqrt{\frac{a}{b^{3}}}\).

Step 4: Simplify expression d

Expression d is \(\sqrt{a}\). This is already simplified and cannot be reduced further.

Step 5: Simplify expression e

Expression e is \(\frac{\sqrt[3]{a^{2}}}{b^{2}}\). This is already simplified and cannot be reduced further.

Step 6: Simplify expression f

Expression f is \(\sqrt[5]{\frac{a^{6} b}{b^{4} \cdot b}}\). Simplify inside the fifth root: \(\sqrt[5]{\frac{a^{6} b}{b^{5}}}\), which simplifies to \(\sqrt[5]{a^{6} b^{1-5}}\) or \(\sqrt[5]{a^{6} b^{-4}}\).

Step 7: Simplify expression g

Expression g is \(\2 a\). This is already simplified and cannot be reduced further.

Step 8: Simplify expression h

Expression h is \(\frac{\sqrt[5]{a^{2} b^{3}}}{\sqrt[5]{b^{5}}}\). Combine the fifth roots in the numerator: \(\sqrt[5]{a^{2} b^{3}} / b\). This simplifies to \(\sqrt[5]{a^{2} b^{-2}}\) or \(\frac{\sqrt[5]{a^{2}}}{b}\).

Key Concepts

fifth roots

A fifth root simplifies expressions by reducing each component under the root symbol to one-fifth of its original exponent. For instance, if you have a number like \(a^5\), the fifth root would be \(a\).
In algebraic expressions, it works similarly. For example, in our exercise, the expression \(\frac{\sqrt[5]{a^{2}} \sqrt[5]{b^{2}}}{\sqrt[5]{b^{5}}}\) can be simplified by combining fifth roots in the numerator: \(\frac{\sqrt[5]{a^{2} b^{2}}}{\sqrt[5]{b^{5}}}\).
We further simplify it as follows:
\[\frac{\sqrt[5]{a^{2}b^{2}}}{b}\]
Fifth roots help simplify complex expressions involving high powers.

fraction simplification

Fraction simplification is crucial in algebra. It helps to reduce fractions to their simplest form.
In our exercise, consider the expression \(\frac{a^{2}}{b^{3}}\). This is already in its simplest form, as there are no common factors to cancel out.
To simplify a fraction, follow these steps:

• Identify common factors in the numerator and denominator.
• Divide both by their greatest common factor.

For example, if we had \(\frac{4a}{8b}\), we would divide both the numerator and the denominator by their greatest common factor, 4, resulting in \(\frac{a}{2b}\).
This method makes algebraic expressions more manageable.

algebraic expressions

Algebraic expressions involve variables and constants combined using operations like addition, subtraction, multiplication, and division. Variables can represent unknown values and are usually denoted by letters like \(a\), \(b\), or \(x\).
In the given exercise, expressions like \(\frac{a^{2}}{b^{3}}\) and \(\sqrt[5]{a^{6} b}{b^{4} \cdot b}\) are examples of algebraic expressions.
To work with these expressions, you often need to:

• Simplify them by combining like terms.
• Factorize where possible.
• Use properties of exponents and roots.

Understanding how to manipulate algebraic expressions is key to solving many math problems.

square roots

Square roots are one of the most common types of roots. They are used to find a number that, when multiplied by itself, yields the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\).
In algebraic expressions, consider the exercise example: \(\sqrt{\frac{a \cdot b}{b^{3} \cdot b}}\).
Simplify inside the square root first:
\[\sqrt{\frac{a \cdot b}{b^{4}}}\]
Then simplify further to:
\[\sqrt{\frac{a}{b^{3}}}\]
Square roots often make expressions simpler by reducing the power of variables, making calculations easier to manage.