Problem 33 - Algebra 2

Question

Simplify. $$ \sqrt{169 x^{8} y^{4}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 13x^4y^2 \).

Step 1: Identify the Expression Components

We are given the expression \( \sqrt{169 x^8 y^4} \). In this expression, 169, \( x^8 \), and \( y^4 \) are under the square root.

Step 2: Simplify the Constant

The number 169 is a perfect square since \( 13^2 = 169 \). Therefore, we can simplify \( \sqrt{169} \) to 13.

Step 3: Simplify the Variable with Exponents - x

We have \( x^8 \) under the square root. The rule of exponents under a square root tells us to divide the exponent by 2. Therefore, \( \sqrt{x^8} = x^{8/2} = x^4 \).

Step 4: Simplify the Variable with Exponents - y

Similarly, we have \( y^4 \) under the square root. By the same rule, \( \sqrt{y^4} = y^{4/2} = y^2 \).

Step 5: Combine the Simplified Components

Putting it all together, we have \( 13 \cdot x^4 \cdot y^2 \). Thus, the simplified form of the expression is \( 13x^4y^2 \).

Key Concepts

Perfect Squares

A perfect square is an integer that can be expressed as the square of another integer. It's called "perfect" because its square root yields a whole number. For instance, 169 is a perfect square because

• When you take the square root of 169, you get 13, since \( 13 \times 13 = 169 \).

Recognizing perfect squares makes simplifying radical expressions much simpler. When simplifying radicands (numbers inside a square root), always look for possibilities to break them into perfect squares first.
This can significantly reduce the complexity of the problem.
Additionally, knowing common perfect squares such as 4, 9, 16, 25, and so on can be very helpful in identifying these quickly.

Exponent Rules

Exponent rules are guidelines that make it easier to work with powers and roots. One of the fundamental rules is that when you take a square root of a number raised to a power, you simply divide the exponent by two.

• For example, \( \sqrt{x^8} \) becomes \( x^{8/2} \) which simplifies to \( x^4 \).
• Similarly, \( \sqrt{y^4} \) becomes \( y^{4/2} = y^2 \).

This works because the exponent essentially tells you how many times to multiply the base by itself.
And dividing the exponent by the root (which is 2 for square roots) adjusts those repetitions accordingly.
Another useful rule is the multiplication of like bases: when you multiply two numbers with the same base, you add the exponents. However, that concept is more advanced and not directly applicable to this exercise.

Variables with Exponents

Variables with exponents indicate how many times the variable is used in the multiplication. Simplifying expressions involving these requires understanding the nature of exponents.

• When a variable like \( x^8 \) is inside a square root, using exponent rules helps to condense the problem.
• Dividing the exponent inside the square root simplifies \( \sqrt{x^8} \) to \( x^4 \).

This method can be extended to other algebraic expressions, where each variable's exponent under the square root is halved.
It's crucial to remember that this is specifically for even exponents under the square root to ensure the resulting exponents are integers.
By practicing with different expressions, you can become adept at quickly simplifying any radical involving variables with exponents.