Simplifying Exponents

Properties of Exponents

There are several properties of exponents which are frequently used to manipulate and simplify algebraic and arithmetic expressions.

${\begin{aligned}x^{m}x^{n}&=x^{m+n}\\{\dfrac {x^{m}}{x^{n}}}&=x^{m-n}\\(xy)^{n}&=x^{n}y^{n}\\\left({\dfrac {x}{y}}\right)^{n}&={\dfrac {x^{n}}{y^{n}}}\\(x^{m})^{n}&=x^{mn}\end{aligned}}$

The first, third, and fifth properties may be extended to several factors, as follows:

${\begin{aligned}x^{m_{1}}x^{m_{2}}x^{m_{3}}\cdots x^{m_{n}}&=x^{m_{1}+m_{2}+m_{3}+\,\cdots \,+m_{n}}\\(x_{1}x_{2}x_{3}\cdots x_{n})^{m}&=x_{1}^{m}x_{2}^{m}x_{3}^{m}\cdots x_{n}^{m}\\(\cdots (((x^{m_{1}})^{m_{2}})^{m_{3}})\cdots )^{m_{n}}&=x^{m_{1}m_{2}m_{3}\cdots \,m_{n}}\end{aligned}}$

Bases and exponents of one and zero

Any number raised to an exponent of one equals itself. So, for example, 5¹ = 5.

Any non-zero number raised to an exponent of zero equals one. So, for example, 5⁰ = 1.

Zero raised to any positive exponent is still zero. So, for example, 0⁵ = 0.

One raised to any exponent is still one. So, for example, 1⁵ = 1.

Zero raised to an exponent of zero is not defined.

Negative and noninteger powers

A base may also be raised to a negative, fractional, or decimal power. These will be covered later in this lesson

Combining powers and roots

The unit fraction notation used for roots previously may have given you the idea that roots are really the same as powers, only with a unit fraction (one over some number) instead of an integer as the exponent. Thus, the fractional notation is actually preferred in higher mathematics, although the root symbol is still used occasionally, especially for square roots.

Fractions as exponents

Other (non-unit) fractions may also be used as exponents. In this case, the base number may be raised to the power of the numerator (top number in the fraction) then the denominator (bottom number) may be used to take the root. For example:

$8^{2/3}={\sqrt[{3}]{8^{2}}}={\sqrt[{3}]{64}}=4$

Alternatively, you can take the root first and then apply the power:

$8^{2/3}=({\sqrt[{3}]{8}})^{2}=2^{2}=4$

Decimal exponents

Any fractional exponent can also be expressed as a decimal exponent. For example, a square root may also be written as:

$25^{1/2}=25^{0.5}=5\;$

Also, decimals which can't be expressed as a fraction (irrational numbers) may be used as exponents:

$5^{3.1415926...}=156.99...\;$

Such problems aren't easy to solve by hand using basic math skills, but the answer can be estimated manually. In this case, since 3.1415926 is between 3 and 4 (and considerable closer to 3), we know that the answer will be between 5^3 (or 125) and 5^4 (or 625), and considerable closer to 125.

Negative exponents

A negative exponent simply means you take the reciprocal (one over the number) of the base first, then apply the exponent:

$25^{-0.5}=1/25^{0.5}=1/5\;$

Alternatively, you can first apply the exponent (ignoring the sign), then take the reciprocal:

$25^{-0.5}=5^{-1}=1/5\;$

Fractions as bases

When a fraction is raised to an exponent, both the numerator and denominator are raised to that exponent:

$(5/6)^{2}=5^{2}/6^{2}=25/36\;$

Fractions may also be used for both the base and exponent:

$(25/36)^{1/2}=25^{1/2}/36^{1/2}=5/6\;$

In addition, negative fractional exponents may be used, taking the reciprocal of the base, as always, to find the solution:

$(25/36)^{-1/2}=(36/25)^{1/2}=36^{1/2}/25^{1/2}=6/5\;$

Negative bases

Negative bases can be handled normally for integer powers:

$(-5)^{2}=(-5)\times (-5)=25\;$

$(-5)^{3}=(-5)\times (-5)\times (-5)=-125\;$

$(-5)^{4}=(-5)\times (-5)\times (-5)\times (-5)=625\;$

$(-5)^{5}=(-5)\times (-5)\times (-5)\times (-5)\times (-5)=-3125\;$

Note that negative bases raised to even powers produce positive results, while negative bases raised to odd powers produce negative results.

Be careful with negative signs. Since -5 = -1×5, there is a difference between $-5^{2}$ and $(-5)^{2}$. The former means the negative of 5 times 5, whereas the latter
means -5 squared. In other words,

$-5^{2}=-1\times 5^{2}=-1\times (5\times 5)=-1\times 25=-25$

but

$(-5)^{2}=(-5)\times (-5)=25$

Roots and fractional/decimal powers are a bit trickier. Odd roots work out fine:

$(-125)^{1/3}=-5\;$

Even roots, however, have no real solution:

$(-25)^{1/2}=?\;$

$(-625)^{1/4}=?\;$

Note that there is no real number, when multiplied by itself, which will produce -25, because 5×5 = 25 and -5×-5 = 25. There is actually a solution, called an imaginary number, but that won't be discussed until later lessons.

Principal root

Note that, since both 5×5 = 25 and -5×-5 = 25, when we are asked to take the square root of 25 there are, in fact, two valid answers, 5 and -5. Actually, any even root of a positive number will have two solutions, with one being the negative of the other. This may seem unusual, but, in higher mathematics, problems often have multiple solutions.

However, for many problems, only the positive value seems to physically work. For example, if we are asked to figure the length of the sides of a square yard which has an area of 25 square units, only 5 units on a side works. If we said "each side can also have a length of -5 units", that doesn't make any sense. For this reason, the positive solution is called the principal root, and, depending on the question, may be the only desired answer. In cases where either answer is valid, it is sometimes written as ±5 (read as "plus or minus five"). However, the mathematical definition of the square root of x squared is the absolute value of x. Thus, square roots equations do not have two answers but two numbers can square to equal the same rational number.

Resources

• Simplifying Exponents, The Organic Chemistry Tutor

Licensing

Content obtained and/or adapted from:

• Powers, roots, and exponents; Wikibooks: Primary Mathematics under a CC BY-SA license