Problem 40 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Answer: $y^2$

Step 1: Start with the given expression

We are given: $$ \sqrt{y^{3}} \sqrt{y} $$

Step 2: Rewrite the square roots using exponents

We can rewrite the square roots using exponents. The square root of a number is equivalent to raising the number to the power of 1/2: $$ (y^{3})^{\frac{1}{2}} (y)^{\frac{1}{2}} $$

Step 3: Apply the power-to-power rule

Now, apply the power-to-power rule to simplify the expression. According to this rule, when we have an exponent raised to another exponent, we multiply the exponents: $$ y^{\frac{3}{2}} y^{\frac{1}{2}} $$

Step 4: Add the exponents when multiplying the same base

Since we are multiplying two terms with the same base (y), we can add the exponents to simplify the expression further: $$ y^{\frac{3}{2} + \frac{1}{2}} $$

Step 5: Simplify the expression by adding the exponents

Finally, we add the exponents: $$ y^{\frac{4}{2}} $$

Step 6: Rewrite the expression as a square root

Now, we can simplify the expression by rewriting it back in the form of a square root: $$ y^{2} $$ So, the product of the given expression is: $$ y^2 $$

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of the same number. For example, when we say \( y^3 \), it means \( y \times y \times y \). The number raised above, which is 3 in this case, is called the exponent, while \( y \) is the base. Understanding exponents allows us to deal with larger numbers efficiently.

• Simple Meaning: An exponent indicates how many times to multiply a number by itself.
• Power Notation: It is written as a small number above the base number. For example, \( 2^4 \) is \( 2 \) multiplied by itself four times.
• Examples: \( 3^2 = 9 \), because \( 3 \times 3 = 9 \).

Exponents are powerful tools in mathematics because they make it easier to write and work with large numbers, simplifying complex calculations into more manageable forms.

Square Roots

Square roots can be thought of as the opposite of squaring a number. If squaring a number means multiplying it by itself, finding the square root means determining which number, when multiplied by itself, gives the original number.

• Symbol: The square root is represented by the radical sign \( \sqrt{ } \).
• Example: \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \).
• Use on Exponents: A square root can be rewritten as an exponent of \( \frac{1}{2} \).

When dealing with algebraic expressions, converting square roots to exponents can simplify multiplication or division of powers involving square roots.

Multiplication of Powers

This concept deals with the rules that simplify the multiplication of expressions that have the same base but different exponents. The rule states that when you multiply powers with the same base, you add the exponents.

• Formula: \( a^m \times a^n = a^{m+n} \).
• Example: \( x^2 \times x^3 = x^{2+3} = x^5 \).
• Why It Works: Since exponents represent repeated multiplication, adding them logically accumulates the total number of times the base is used as a factor.

This rule simplifies complex problems by reducing multiple multiplications into a simple addition of exponents, making calculations quicker and errors less likely.