Problem 46 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{y^{7}} \sqrt{y^{9}} $$

Step-by-Step Solution

Answer

Question: Find the product of the two square root expressions: $$\sqrt{y^7}\sqrt{y^9}$$ Answer: The product of the given expressions is $$y^8$$.

Step 1: Rewrite the expression with a single radical

We rewrite the two-square-root expression into a single square root with the product of their radicands: $$ \sqrt{y^7} \cdot \sqrt{y^9} = \sqrt{y^7 \cdot y^9} $$

Step 2: Use the properties of exponents

Now we will simplify the expression inside the square root using the property of exponents, which states that \(a^m \cdot a^n = a^{m+n}\). In this case, \(a = y\), \(m = 7\), and \(n = 9\): $$ \sqrt{y^7 \cdot y^9} = \sqrt{y^{7+9}} $$

Step 3: Compute the sum of the exponents

Calculate the sum of the exponents: $$ \sqrt{y^{7+9}} = \sqrt{y^{16}} $$

Step 4: Simplify the square root

Finally, we will simplify the square root using our knowledge of roots and exponents. The square root of a variable raised to an even power is the variable raised to half of that power: $$ \sqrt{y^{16}} = y^8 $$ Thus, the product of the given expressions is: $$ \sqrt{y^7}\sqrt{y^9} = y^8 $$

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial when simplifying mathematical expressions that involve exponential terms. The key property used in the exercise is known as the Product of Powers, which states that when multiplying two expressions with the same base, one can add the exponents. For example, given two variables such as \( a^m \) and \( a^n \), their product is \( a^{m+n} \).

This property becomes especially handy when dealing with square roots of exponential terms. With our problem \( \sqrt{y^7} \cdot \sqrt{y^9} \), the base is \( y \), and by applying the Product of Powers property, we combine the exponents, transforming it into \( \sqrt{y^{16}} \). By recognizing these properties, students can streamline the simplification process without distributing the exponents individually to each factor under the radical.

Radicals Multiplication

When we multiply radicals, there are certain rules to follow for correct simplification. It's imperative to recall that the product of two square roots can be combined into one square root that encompasses the product of the radicands (the number or expression inside the radical). Essentially, \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \), given that 'a' and 'b' are non-negative, as negative numbers would involve complex numbers.

In our example, \( \sqrt{y^7} \cdot \sqrt{y^9} \) is simplified to \( \sqrt{y^7 \cdot y^9} \). This ease of multiplication under a single radical makes it simpler to apply exponent rules. It is important to keep these guidelines in mind as missteps in radical multiplication can lead to incorrect answers.

Exponent Rules

Exponent rules are a group of guidelines that describe how to handle mathematical operations involving exponents. One critical rule, as discussed earlier, is the Product of Powers. There are others, such as the Power of a Power, where \( (a^m)^n = a^{m \cdot n} \), and the Quotient of Powers, which gives us \( a^m / a^n = a^{m-n} \) when \( m \) and \( n \) are integers and \( a \) is non-zero.

In simplifying the square root of \( y^{16} \), we applied another rule: the square root of a power. This rule asserts that \( \sqrt{a^m} = a^{m/2} \) when 'm' is an even number. As a result, \( \sqrt{y^{16}} = y^8 \). These rules ensure that complicated expressions are dealt with consistently and accurately, providing clarity in both understanding and computation for students.