Problem 42 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \sqrt{x^{12} y^{10} z^{8} w^{7}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\sqrt{x^{12}y^{10}z^{8}w^{7}}\). Answer: \(x^6 y^5 z^4 \sqrt{w^7}\)

Step 1: Write down the given expression

We have the expression: $$ \sqrt{x^{12}y^{10}z^{8}w^{7}} $$

Step 2: Apply the square root property

Using the property of square roots, we get: $$ \sqrt{x^{12}y^{10}z^{8}w^{7}} = x^{\frac{12}{2}}y^{\frac{10}{2}}z^{\frac{8}{2}}\sqrt{w^7} $$

Step 3: Simplify the exponents

Now we simplify the exponents: $$ x^6 y^5 z^4 \sqrt{w^7} $$ That's the simplified result. The final answer is: $$ x^6 y^5 z^4 \sqrt{w^7} $$

Key Concepts

Square Root

The concept of a square root involves finding a number that, when multiplied by itself, gives the original number. In mathematics, the square root is often represented as \( \sqrt{} \). For example, the square root of 9 is 3, because \( 3 \times 3 = 9 \).

In the context of expressions, the square root function can also be applied to variables and expressions. When we have an expression like \( \sqrt{x^{12}y^{10}z^{8}w^{7}} \), we are looking to simplify it by identifying pairs of powers, because taking the square root of a perfect square results in an integer.

• The square root property states that \( \sqrt{a^2} = a \).
• This principle can be applied to higher powers as well, such as \( \sqrt{x^{12}} = x^{12/2} = x^6 \).

Exponents

Exponents, small numbers positioned at the upper right of a base number or variable, indicate how many times the base is multiplied by itself. For example, in the expression \( x^{12} \), 12 is the exponent which signifies that \( x \) is multiplied 12 times.

When simplifying square root expressions with exponents involved, the properties of exponents can be very helpful. Specifically, the rule that \( a^{m/n} = \sqrt[n]{a^m} \) is important, where \( m \) is the original exponent and \( n \) is the root.

• Divide the exponent by 2 when simplifying square roots: \( x^{12} \) becomes \( x^{12/2} = x^6 \).
• This division is seen in the expression: \( \sqrt{x^{12}y^{10}z^{8}w^{7}} = x^{6}y^{5}z^{4}\sqrt{w^7} \).

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. The radical symbol \( \sqrt{} \) is used to denote square roots.

To simplify a radical expression, you should aim to pull out any perfect squares from under the radical sign. This makes the expression simpler and easier to handle. In our expression \( \sqrt{x^{12}y^{10}z^{8}w^{7}} \), most of the components can be simplified completely out of the radical.

• Pull out pairs from under the square root: each pair of exponents becomes a whole number.
• For example, \( w^7 \) becomes \( w^3\sqrt{w} \) after extracting pairs.

Understanding how to manipulate radical expressions helps make algebraic problems more tractable, allowing you to solve them more efficiently.