Some algebraic expressions require multiplying or dividing exponential terms. Rather than computing each exponential term and multiplying or dividing manually, simply add exponents when multiplying and subtract when dividing to save time. This concept can also be used to simplify variable expressions. 

\(\displaystyle 5x^{3} (7x^{2}) = (5x7) x^{3+2} = 35x^{5}\)

\(\displaystyle \frac{x^{12}}{x^{9}} = x^{12-9} = x^{3}\)

\(\displaystyle 35x^{5} + x^{3}\)  is the correct answer.

To simplify this algebraic expression, apply the distributive property:

\(\displaystyle \frac{16x^{2}+ 24x-4}{4} = \frac{16^{2}}{4} + \frac{24x}{4} - \frac{4}{4}\)

\(\displaystyle 4x^{2} + 24x -1\) is the correct answer.

To simplify this algebraic expression, factor the numerator and the denominator using FOIL.

\(\displaystyle \frac{(x-8)(x+1)}{(x+2) (x+1)}\)

The \(\displaystyle \frac{x+1}{x+1}\) will cancel out.

The correct answer is

\(\displaystyle \frac{x-8}{x+2}\)

To simplify this algebraic expression, factor the denominator using the FOIL method.

\(\displaystyle x^{2} + 2x - 15 = (x+5) (x-3)\)

\(\displaystyle \frac{x+5}{(x+5) (x-3)}\)

The \(\displaystyle \frac{x+5}{x+5} = 1\)

Solve:

\(\displaystyle \frac{1}{x-3}\)

To simplify, factor the numerator using the FOIL Method:

\(\displaystyle 9x^{2} -6x - 3 = (3x-3) (3x+1)\)

Rewrite the expression:

\(\displaystyle \frac{(3x-3) (3x+1)}{(3x+1)}\)

\(\displaystyle \frac{3x-3}{1} = 3x-3\)

To reduce the expression, you need a common factor to take out from each part of the equation.  

All the constants can be divided by \(\displaystyle 2\) so you just divide the whole equation by that to get,

\(\displaystyle \\4x-6y=8 \\ \\2(2x-3y)=2(4) \\ \\\frac{2(2x-3y)}{2}=\frac{2(4)}{2}\)

\(\displaystyle 2x-3y=4\).

To solve this equation, begin by using the Order of Operations.  The first operation will be the one that is contained within the parentheses.  When a term with an exponent is raised to a power, multiply the exponents.

\(\displaystyle (x^{2})^{3} = x^{6}\)

Rewrite the expression:

\(\displaystyle 18 + 6y + 20y + 12 - x^{6}\)

Simplify by combining like terms:

\(\displaystyle 30 + 26y -x^{6}\)

There is one more step. The terms must be placed in descending order based upon the exponents. In this equation, the term \(\displaystyle -x^{6}\) , has the largest exponent so it will go first followed by \(\displaystyle 26y\) which has an exponent of 1.  Constants are placed last in the equation.

\(\displaystyle -x^{6} + 26y + 30\)

To simplify this algebraic expression, combine like terms. Make sure to distribute the minus sin to all terms and constants within the parentheses. 

\(\displaystyle 6m^{2} n^{3} - (-3m^{2}n^{3}) = 9m^{2}n^{3}\)

\(\displaystyle 9m^{2}n^{3} -6mn - (+3) -6\)

\(\displaystyle 9m^{2}n^{3} -6mn -3 - 6\)

\(\displaystyle 9m^{2}n^{3} - 6mn -9\) is the correct answer.

\(\displaystyle -45 + 8 (x +6)\)

When solving this problem we need to remember our order of operations, or PEMDAS. 

PEMDAS stands for parentheses, exponents, multiplication/division, and addition/subtraction. When you have a problem with several different operations, you need to solve the problem in this order and you work from left to right for multiplication/division and addition/subtraction.

Parentheses: We are not able to add a variable  to a number, so we move to the next step. 

Multiplication: We can distribute (or multiply) the \(\displaystyle 8\). 

\(\displaystyle =-45 + 8 \cdot x +8 \cdot6\)

\(\displaystyle =-45 + 8x +48\)

Addition/Subtraction: Remember, we can't add a variable  to a number, so the \(\displaystyle 8x\) is left alone.

Now we have \(\displaystyle -45+48=3\)

\(\displaystyle = 8x + 3\)

\(\displaystyle 30 - 5 (x + 4)\)

When solving this problem we need to remember our order of operations, or PEMDAS. 

PEMDAS stands for parentheses, exponents, multiplication/division, and addition/subtraction. When you have a problem with several different operations, you need to solve the problem in this order and you work from left to right for multiplication/division and addition/subtraction.

Parentheses: We are not able to add a variable  to a number, so we move to the next step. 

Multiplication: We can distribute (or multiply) the \(\displaystyle -5\). 

\(\displaystyle = 30 - 5 \cdot x + (- 5) \cdot 4\)

\(\displaystyle = 30 - 5 x + (- 20)\)

Addition/Subtraction: Remember, we can't add a variable  to a number, so the \(\displaystyle -5x\) is left alone.

\(\displaystyle = - 5 x + 30 - 20\)

\(\displaystyle = -5x + 10\)