Problem 59 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \frac{\left(2 x^{3} y^{3}\right)^{4}\left(5 x^{6} y^{8}\right)^{2}}{\left(4 x^{5} y^{3}\right)^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $$\frac{ (2^4 x^3 y^3)^4 (5^2 x^6 y^8)^2}{(4^2 x^5 y^3)^2}$$ Answer: The simplified expression is $$25 x^{14} y^{22}$$

Step 1: Expand the expression using Power Rules

First, we will expand the expression by applying the power rule \((a b)^{n}=a^{n} b^{n}\) to each term: $$ \frac{\left( (2^4)x^{(3\cdot4)} y^{(3\cdot4)}\right)\left((5^2)x^{(6\cdot2)} y^{(8\cdot2)}\right)}{\left((4^2) x^{(5\cdot2)} y^{(3\cdot2)}\right)} $$

Step 2: Calculate the powers of constants and variables

Now, we will compute the powers of constants and variables for each term: $$ \frac{ (16) x^{12} y^{12} (25)x^{12}y^{16}} { (16) x^{10} y^{6}} $$

Step 3: Multiply the coefficients and variables on the numerator

In this step, we will multiply the coefficients and combine the like terms for variables: $$ \frac{ (16\cdot25) x^{(12+12)} y^{(12+16)}}{ (16) x^{10} y^{6}} $$ $$ \frac{ (400) x^{24} y^{28}}{ (16) x^{10} y^{6}} $$

Step 4: Simplify the fractions and expressions

Finally, we will simplify the fractions and expressions by canceling out the similar terms in the numerator and denominator: $$ \frac{ (400) x^{24} y^{28}}{ (16) x^{10} y^{6}} = \frac{ (25) x^ {(24-10)} y^{(28-6)}}{ x^{10} y^{6}} $$ To do this, divide both the numerator and the denominator by their common factors: $$ \frac{(400\div16)x^{(24-10)}y^{(28-6)}}{x^{10}y^{6}}=\frac{25 x^{14}y^{22}}{x^{10}y^{6}} $$ So, the simplified expression is: $$ 25 x^{14} y^{22} $$

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is all about making a complex expression easier to understand and work with. To simplify an expression, follow the rules of arithmetic and algebra to reduce it to a more manageable form.
For instance, when multiplying powers with the same base, we add the exponents. If we have

• \(a^m \times a^n = a^{m+n}\).

When dividing, we subtract the exponents of like bases:

• \(\frac{a^m}{a^n} = a^{m-n}\).

You can simplify by canceling out common factors in numerators and denominators. This is a powerful technique for breaking down expressions into their simplest form.
Look at terms carefully to see if they can be combined or simplified using these basic rules.

Natural Numbers

Natural numbers are a fundamental concept in mathematics. They are the numbers you count with, like 1, 2, 3, and so forth. When dealing with exponents, natural numbers tell you how many times to multiply the base by itself. Consider an example

• \(a^n\) where \(a\) is a base, and \(n\) is a natural number.

These numbers are always positive, starting from 1 and increasing to infinity. When simplifying expressions involving exponents, understanding that they consist of natural numbers helps in applying rules correctly.
They make recognizing patterns and calculating powers straightforward, as exponents only use positive integer values.

Variable Exponents

Variable exponents involve situations where the exponent itself is a variable, such as \(x^m\) where both the base and the power can change. Handling variable exponents can be more complex, as it requires applying laws of exponents to make the expression easier to manage.
For example, when both the base and the power are variables, you rely on the same power rules.

• When multiplying like bases: \(x^a \times x^b = x^{a+b}\).


• For division: \(\frac{x^a}{x^b} = x^{a-b}\).

It's crucial to keep the results in terms of simplified exponents, making calculations and further algebraic manipulations easier. This step of reducing exponents can lead to more convenient solutions in equations and algebraic fractions.

Algebraic Fractions

Algebraic fractions are ratios of two polynomial expressions, just like regular fractions but with variables. Simplifying algebraic fractions involves reducing the terms using techniques similar to simplifying numeric fractions and expressions.
Look for common factors in both the numerator and the denominator and cancel them out.

• This clears out redundant parts of the expression, making it easier to handle.

For instance, in an expression like

• \(\frac{x^5y^3}{x^3y^2}\) you can apply exponent rules and simplify it to \(x^{5-3}y^{3-2} = x^2y\).

Simplifying algebraic fractions is crucial for solving complex equations, as it can transform them into more straightforward, workable forms. By reducing expressions to their simplest form, you can better analyze and solve mathematical problems.