Math Homework Help Exponents

Overview

Exponents are used as a type of shorthand directing how many times that a base number is multiplied by itself. Exponential expressions are useful when working with very large or very small numbers, as in scientific notation.   Like other numbers and variables, exponential expressions can be multiplied following certain rules.

Review of Simplifying Expressions

If there are negative exponents in an expression, it is not simplified. The expression x-3 can be defined as 1/x3.  Similarly, if the same base is used more than once in an expression, it is not simplified.  Suppose the expression is x2x.  That actually means x·x·x, or x3.  Also, powers that are raised to powers are not simplified, such as [x5]2.  If any coefficients have common factors, those common factors must be simplified.  For example, (15a2)/ (10b3) can be simplified to (3a2)/ (2b3) because 15 and 10 have a common factor of 5.

 

Products of Powers

Since an exponent stands for how many times a base is multiplied by itself, the rule for multiplying exponents can be discovered by showing each power as repeated multiplication.  If the expression is y3y2, that can be expanded as y·y·y·y·y, or y5.  The product of the powers can be found by adding the exponents, as 3 +2 = 5.  However, the bases must be the same in order for the exponents to be added. In the language of algebra, for any real number a not equal to zero, am·an = am+n.

 

Powers of Powers

Expressions such as [x5]2 actually mean x5·x5, or (x·x·x·x·x) (x·x·x·x·x), by using the meaning of exponents as repeated multiplication.  The product of powers of powers can be found by multiplying the exponents, as 5·2 equals 10.  In the language of algebra, for any real number a not equal to zero, (am)n=amn.  The way to know whether to add or multiply the exponents is to expand the expression to repeated multiplication.  For example, z3z4 means z·z·z·z·z·z·z, or z7, while (z3)4 means (z3)(z3)(z3)(z3), or (z·z·z)(z·z·z)(z·z·z)(z·z·z) or z12.

Powers of Products

An expression such as q5 only involves one component, the variable q.  However, an expression such as (3z)3 has two components, the coefficient 3 and the variable z.  It can be expanded as (3z)(3z)(3z), or 3·3·3·z·z·z, or 33z3, or 27z3.  In the language of algebra, (ab)n, for any real number a and b not equal to zero and n as an integer, (ab)n equals anbn.

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What is an Exponent?

Quick Explanation:

An exponent is a short-handed method of expressing repeated multiplication. Rather than writing \(5*5\) we can simply write \(5^2\). They mean the same thing -- the superscript 2 means to multiply five twice. Similarly, \(y^4\) means multiply y four times, i.e. \(y*y*y*y\).

More Detail:

It doesn't seem all that hard to just write \(5*5\) instead of \(5^2\), but there are cases where the exponent could be quite large -- imagine writing out \(5^{25}\)! Farther down the road you'll also see that exponents can be negative, and don't even have to be whole numbers! There is far more to the exponent than simply saving time writing out multiplication. You'll also learn that many, many functions in math and in nature have a squared dependence -- that means one quantity depends on another value squared, or raised to the second power. For example, the area of a square is a function of the side length squared. Believe it or not, exponents will make things easier... just have patience!

So, how can I work with exponents?

If you understand that an exponent represents the number of times you multiply something, you can immediately understand what happens when we multiply two variables with exponents:

Example

Simplify this expression: \(x^2*x^6\)

Since \(x^2\) really just means \(x*x\), and \(x^6\) just means 6 more x's, we end up with 8 x's multiplied together, right? Well what is an exponent -- the number of times we multiply something! Therefore, \(x^2*x^6\) equals \(x^8\).

What did you learn from that example? When the same variable is multiplied, any exponents are added together. Adding to the exponent is the same as multiplying more times:

Rule 1: \( x^a*x^b=x^{a+b}\)

There are other rules with exponents as well. If multiplying two variables adds their exponents, then division must subtract exponents! Check out this example:

Simplify: \( \large \frac{x^6}{x^3} \)

Well, remember that this is just a quick way of writing the following:

\( \large \frac{x*x*x*x*x*x}{x*x*x} \)

Hopefully you remember enough basic algebra that you know to cancel a factor that's in the numerator and the denominator. In fact, we can scratch off three of the x's, leaving just the numerator:

\( x*x*x=x^3 \)

So that gives us another rule:

Rule 2: \( \frac{x^a}{x^b}=x^{a-b} \)

Let's introduce a few more rules of exponents quickly:

Rule 3: \( x^1=x\)

That rule makes sense, because having just one x can't equal anything else but x, right? The next one might make a little less sense, but here it is:

Rule 4: \(x^0=1\)

That's right - anything raised to the 0 power is 1. Here's why this is the case. With exponents, we are working with multiplication. The identity in multiplication is 1. Imagine taking rule 1, and adding in a couple 0's:

$$ x^a*x^b = x^{a+b+0+0} $$

Logically, we shouldn't have changed anything by adding those zeros. Since you can rewrite it as:

$$ x^a*x^b*x^0*x^0 $$

In order for that to be the same as \(x^a*x^b\), the \(x^0\) factors must equal 1.

 

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