A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in [link], and use the Pythagorean Theorem.
Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.
When the square root of a number is squared, the result is the original number. Sincethe square root ofisThe square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.
In general terms, ifis a positive real number, then the square root ofis a number that, when multiplied by itself, givesThe square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equalsThe square root obtained using a calculator is the principal square root.
The principal square root ofis written asThe symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.
The principal square root ofis the nonnegative number that, when multiplied by itself, equalsIt is written as a radical expression, with a symbol called a radical over the term called the radicand:
Does
No. Although bothandarethe radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is
Evaluate each expression.
Forcan we find the square roots before adding?
No.This is not equivalent toThe order of operations requires us to add the terms in the radicand before finding the square root.
Evaluate each expression.
To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewriteasWe can also use the product rule to express the product of multiple radical expressions as a single radical expression.
Ifandare nonnegative, the square root of the productis equal to the product of the square roots ofand
Given a square root radical expression, use the product rule to simplify it.
Simplify the radical expression.
Simplify
Notice the absolute value signs around x and y? That’s because their value must be positive!
Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.
Simplify the radical expression.
Simplifyassuming
Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewriteas
The square root of the quotientis equal to the quotient of the square roots ofandwhere
Given a radical expression, use the quotient rule to simplify it.
Simplify the radical expression.
Simplify
We do not need the absolute value signs forbecause that term will always be nonnegative.
Simplify the radical expression.
Simplify
We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum ofandisHowever, it is often possible to simplify radical expressions, and that may change the radicand. The radical expressioncan be written with ain the radicand, asso
Given a radical expression requiring addition or subtraction of square roots, solve.
Add
We can rewriteasAccording the product rule, this becomesThe square root ofis 2, so the expression becomeswhich isNow we can the terms have the same radicand so we can add.
Add
Subtract
Rewrite each term so they have equal radicands.
Now the terms have the same radicand so we can subtract.
Subtract
When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.
We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.
For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator ismultiply by
For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator isthen the conjugate is
Given an expression with a single square root radical term in the denominator, rationalize the denominator.
Writein simplest form.
The radical in the denominator isSo multiply the fraction byThen simplify.
Writein simplest form.
Given an expression with a radical term and a constant in the denominator, rationalize the denominator.
Writein simplest form.
Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate ofisThen multiply the fraction by
Writein simplest form.
Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.
Suppose we know thatWe want to find what number raised to the 3rd power is equal to 8. Sincewe say that 2 is the cube root of 8.
The nth root ofis a number that, when raised to the nth power, givesFor example,is the 5th root ofbecauseIfis a real number with at least one nth root, then the principal nth root ofis the number with the same sign asthat, when raised to the nth power, equals
The principal nth root ofis written aswhereis a positive integer greater than or equal to 2. In the radical expression,is called the index of the radical.
Ifis a real number with at least one nth root, then the principal nth root ofwritten asis the number with the same sign asthat, when raised to the nth power, equalsThe index of the radical is
Simplify each of the following:
Simplify.
Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the indexis even, thencannot be negative.
We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.
Rational exponents are another way to express principal nth roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is
Given an expression with a rational exponent, write the expression as a radical.
Writeas a radical. Simplify.
The 2 tells us the power and the 3 tells us the root.
We know thatbecauseBecause the cube root is easy to find, it is easiest to find the cube root before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.
Writeas a radical. Simplify.
Writeusing a rational exponent.
The power is 2 and the root is 7, so the rational exponent will beWe getUsing properties of exponents, we get
Writeusing a rational exponent.
Simplify:
Simplify
Access these online resources for additional instruction and practice with radicals and rational exponents.
What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.
When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.
Where would radicals come in the order of operations? Explain why.
Every number will have two square roots. What is the principal square root?
The principal square root is the nonnegative root of the number.
Can a radical with a negative radicand have a real square root? Why or why not?
For the following exercises, simplify each expression.
16
10
14
25
For the following exercises, simplify each expression.
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluatingWhat is the length of the guy wire?
500 feet
A car accelerates at a rate ofwhere t is the time in seconds after the car moves from rest. Simplify the expression.
For the following exercises, simplify each expression.