Problem 73 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{49 x^{2} y^{5} z^{9}}{25 a^{3} b^{11}}} $$

Step-by-Step Solution

Answer

Short Answer: $\frac{7xy^2z^4\sqrt{yz}}{5a^{3/2}b^{11/2}}$

Step 1: Rewrite the square root expression

In the first step, rewrite the given expression under the square root using separate square roots for the numerator and denominator. $$ \sqrt{\frac{49 x^{2} y^{5} z^{9}}{25 a^{3} b^{11}}} = \frac{\sqrt{49 x^{2} y^{5} z^{9}}}{\sqrt{25 a^{3} b^{11}}} $$

Step 2: Find the square root of the individual terms

Now, find the square root of each of the individual terms in the expression. It is easier to find the square root of perfect squares. $$ \frac{\sqrt{49 x^{2} y^{5} z^{9}}}{\sqrt{25 a^{3} b^{11}}} = \frac{7x\sqrt{y^5z^9}}{5ab^5\sqrt{ab}} $$

Step 3: Simplify the remaining square roots

Separate each term under the remaining square roots and rewrite the powers so that they can be simplified easily by using properties of exponents. $$ \frac{7x\sqrt{y^5z^9}}{5ab^5\sqrt{ab}} = \frac{7x\sqrt{y^{2\cdot2+1}z^{2\cdot4+1}}}{5ab^5\sqrt{a^1b^1}} $$ Apply the properties of exponents and square roots to further simplify: $$ \frac{7x\sqrt{y^5z^9}}{5ab^5\sqrt{ab}} = \frac{7x(y^2z^4\sqrt{yz})}{5ab^5(a^{1/2} b^{1/2})} $$

Step 4: Simplify the final expression

Combine and simplify the terms in the numerator and denominator of the final expression. $$ \frac{7x\sqrt{y^5z^9}}{5ab^5\sqrt{ab}} = \frac{7xy^2z^4\sqrt{yz}}{5a^{3/2}b^{11/2}} $$ The simplified radical expression is: $$ \frac{7xy^2z^4\sqrt{yz}}{5a^{3/2}b^{11/2}} $$

Key Concepts

Square Roots

Understanding square roots is crucial for simplifying radical expressions. A square root asks you to find a number that, when multiplied by itself, gives you the original number. The square root of 4, for example, is 2 because when you multiply 2 by itself (\(2 \times 2\text{ or }2^2\text{)}, you get 4.

In algebra, we often deal with square roots that involve variables and exponents. When simplifying expressions like \( \sqrt{x^2} \), we find that the simplified form is simply \( x \), assuming that \( x \geq 0 \) since we take the principal or non-negative square root. For the given exercise, we saw that \( \sqrt{49} = 7 \) and \( \sqrt{x^{2}} = x \) since 49 is a perfect square and \( x^{2} \) has an exponent that is a multiple of 2, which is the index of the square root.

However, the process becomes more complex when dealing with imperfect squares and higher powers, as we need to manipulate the expression using various techniques to separate the perfect squares from those terms that are not.

Properties of Exponents

The properties of exponents are essential tools when working to simplify radical expressions. These properties are rules that tell us how numbers raised to powers behave under different mathematical operations, such as multiplication, division, and taking roots.

Some key exponent rules that are particularly useful in radical simplification include:

• The Product of Powers rule: \( a^m \times a^n = a^{m+n} \) where you add exponents when multiplying like bases.
• The Power of a Product rule: \( (ab)^n = a^n \times b^n \) which allows you to distribute the exponent to each factor in a product.
• The Power of a Power rule: \( (a^m)^n = a^{m \times n} \) where the exponents are multiplied for a power raised to another power.

In the provided exercise, we use these properties to break down the exponents under the radical so that they are in a form that can be easily simplified. This includes expressing exponents that allow us to take out square roots directly, such as rewriting \( y^5 \) as \( y^{2 \times 2 + 1} \) which simplifies to \( y^2 \sqrt{y} \) when you apply the square root.

Radical Simplification

Radical simplification involves reducing the complexity of expressions that contain roots. The goal is to express the radical in its simplest form, which often means removing any perfect squares from under the root and simplifying any coefficients or terms outside the root.

In the context of our exercise, \( y^5 \) and \( z^9 \) are initially under the square root. By expressing these powers in terms of even numbers, which signify the number of pairs, and odd numbers, we are able to root out the pairs and leave the single 'unpaired' variables under the root.

After breaking down \( y^5 \) as \( y^{2\times2+1} \) and taking \( y^2 \) out of the radical, we're left with \( y \sqrt{y} \). The process is called 'simplifying the radical' because you're simplifying the expression to a form where the radical part is as small as possible.