Problem 43 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{a^{3}} \sqrt{a^{5}} $$

Step-by-Step Solution

Answer

Question: Multiply and simplify the square roots of the following expressions: \(sqrt(a^3)\) and \(sqrt(a^5)\). Answer: \(a^4\)

Step 1: Rewrite the square roots as expressions with exponents

Recall the rule, \(\sqrt{x^n} = x^{\frac{n}{2}}\). So, rewrite the given expression as: $$ a^{\frac{3}{2}} \cdot a^{\frac{5}{2}} $$

Step 2: Apply the product rule for exponents

The product rule for exponents states that when multiplying two expressions with the same base and different exponents, we add the exponents: \(a^m \cdot a^n = a^{m+n}\). Apply the product rule to the expression: $$ a^{\frac{3}{2}+\frac{5}{2}} $$

Step 3: Simplify the exponent and the expression

Adding the exponents, we get: $$ a^{\frac{3}{2}+\frac{5}{2}} = a^{\frac{8}{2}} $$ Now, rewrite the expression in radical form: $$ a^{\frac{8}{2}} = \sqrt{a^8} $$ Since \(8=2\cdot4\), the square root of \(a^8\) is \(a^4\). So, the final answer is: $$ a^4 $$

Key Concepts

Product Rule for Exponents

When simplifying square roots or expressions with exponents, a fundamental rule comes into play: the product rule for exponents. This rule is vital for algebraic simplification and states that when you multiply two expressions with the same base, you can add their exponents together. In formal terms, if you have a base 'a', with two different exponents 'm' and 'n', the product is still a power of 'a', but with a new exponent which is the sum of the two original ones, written as:
\[a^m \cdot a^n = a^{m+n}\].
In practice, for example, if you multiply \(a^2\) by \(a^3\), using the product rule, you get \(a^{2+3} = a^5\), not \(a^6\). This is a common misconception. The rule ensures that expressions with exponents can be easily simplified without expanding them fully, which is especially useful for higher powers of variables.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher. In algebra, we often work with square roots, denoted by the symbol '\(\sqrt{}\)'. To simplify these expressions, one can convert radicals to expressions with fractional exponents where the numerator indicates the power and the denominator indicates the type of root. For instance, the square root of \(x\) can be expressed as \(x^{\frac{1}{2}}\) and the cube root of \(y\) can be written as \(y^{\frac{1}{3}}\).
Understanding how to work with radical expressions is crucial as they often appear in equations and formulae across various branches of mathematics. In many cases, converting a radical to exponential form allows you to apply the rules of exponents, such as the product rule, in order to simplify the expression.

Algebraic Operations

Algebraic operations, such as addition, subtraction, multiplication, and division, are performed on algebraic expressions to simplify or manipulate them. In the context of square roots and exponents, we focus on multiplication and exponentiation. When combining like terms (terms with the same variable and exponent), we add the coefficients. If dealing with multiplication of variables, we apply the product rule.
For division, when the bases are the same, the exponents are subtracted. Exponentiation has its own set of rules, including the power of a power rule, which states that \((a^m)^n = a^{mn}\), and the product to a power rule, which says that for a product \((ab)^n = a^n \cdot b^n\).
These algebraic operations work together to simplify expressions and solve equations. It's essential to practice these rules to become proficient in algebra, as they form the foundation of more advanced mathematical concepts.