Problem 53 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \sqrt{(y-3 z)^{12}(y+3 z)^{10}(y-5 z)^{3}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \((y-3 z)^{6}(y+3 z)^{5}(y-5 z)^{\frac{3}{2}}\).

Step 1: Observe the given expression

We are given the expression: $$ \sqrt{(y-3 z)^{12}(y+3 z)^{10}(y-5 z)^{3}} $$ Our goal is to simplify this expression.

Step 2: Apply properties of exponents

First, let's use the property of exponents that states that \(\sqrt{a^{n}} = a^{\frac{n}{2}}\). Applying it to our expression, we get: $$ [(y-3 z)^{6}(y+3 z)^{5}(y-5 z)^{\frac{3}{2}}] $$

Step 3: Identify common factors

At this point, we don't see any common factors between the three terms that will allow us to simplify the expression further.

Step 4: Write the simplified expression

After applying the properties of exponents and identifying the common factors, the simplified expression is: $$ (y-3 z)^{6}(y+3 z)^{5}(y-5 z)^{\frac{3}{2}} $$

Key Concepts

Properties of Exponents

Exponents are a way to express repeated multiplication concisely. Understanding the properties of exponents can greatly simplify the process of solving algebraic expressions. Here are a few key properties:

• Product of powers: When multiplying two expressions with the same base, keep the base and add the exponents: \(a^m \times a^n = a^{m+n}\).


• Power of a power: When raising an exponent to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).


• Power of a product: When raising a product to an exponent, distribute the exponent to each factor: \((ab)^n = a^n \times b^n\).


• Square roots and exponents: The square root of a term with an exponent can be rewritten using fractional exponents: \(\sqrt{a^n} = a^{\frac{n}{2}}\).

In the original exercise, this property was used to transform \(\sqrt{(y-3z)^{12}}\) into \((y-3z)^6\), allowing for a simpler expression.

Radicals

Radicals refer to mathematical symbols indicating roots, such as square roots or cube roots. A radical expression can often be simplified by using exponent rules. The square root, the most common radical, can be rewritten using fractional exponents as discussed earlier.
Simplifying radicals involves finding factors that are perfect squares (or perfect cubes for cube roots) and reducing them. For example, \(\sqrt{x^4} = x^2\). When dealing with radical expressions:

• Firstly, try to express the radical in terms of exponents for simplification.


• Look for factors under the radical that are perfect squares or higher powers.


• Apply the properties of exponents, especially the rule for square roots: \(\sqrt{a^n} = a^{\frac{n}{2}}\). This helps in expressing the radical in a simpler form.

In the given exercise, the use of radicals is converted to fractional exponents, making the calculations more direct and manageable.

Algebraic Expressions

At the core of solving the given exercise lies the proper handling of algebraic expressions. These expressions consist of variables, coefficients, and the operations of addition, subtraction, multiplication, and division. When simplifying algebraic expressions, adhere to the following steps:

• Identify like terms, which are terms that have the same variables raised to the same powers, and combine them.


• Use the distributive property to eliminate parentheses by multiplying each term inside the parentheses by the term outside.


• Apply the properties of exponents to simplify expressions with powers and roots.

In the given problem, three different terms \((y-3z)^{12}\), \((y+3z)^{10}\), and \((y-5z)^3\) were simplified. No like terms existed inside the radical, but the expressions were simplified using the properties of exponents. The final step involved rewriting the algebraic expression without a radical, leading to a concise form similar to \((y-3z)^6(y+3z)^5(y-5z)^{\frac{3}{2}}\). Properly understanding and applying these principles can make simplification tasks straightforward.