Lesson Explainer: nth Roots: Expressions and Equations | Nagwa Lesson Explainer: nth Roots: Expressions and Equations | Nagwa

Lesson Explainer: nth Roots: Expressions and Equations Mathematics • Second Year of Secondary School

In this explainer, we will learn how to simplify algebraic expressions and solve algebraic equations involving 𝑛th roots, where 𝑛 is a positive integer greater than or equal to 2.

The 𝑛th root is an important mathematical operation that describes the inverse of a power operation with an exponent 𝑛. Let’s begin by providing a formal definition for the 𝑛th root.

Definition: 𝑛th Roots

The 𝑛th root of a number 𝑥, where 𝑛 is a positive integer, is a number that, when raised to the 𝑛th power, gives 𝑥. We can define this number as 𝑦, such that 𝑥=𝑦.

The 𝑛th root is given by 𝑦=𝑥.

Note that the 𝑛th root of 𝑥 can equivalently be written as 𝑥. It is outside the scope of this lesson to express 𝑛th roots using this notation, but if you already are familiar with the notation, it can act as a helpful tool to understanding how the rules of exponents can be applied to expressions involving roots.

Theorem: Properties of 𝑛th Roots

  • When 𝑎 and 𝑏 are well defined and are real numbers, then 𝑎𝑏 is also defined such that 𝑎𝑏=𝑎𝑏.
  • If 𝑏0, then it is also the case that 𝑎𝑏=𝑎𝑏.
  • When 𝑛 is an odd integer, 𝑎=𝑎=𝑎.
  • When 𝑛 is an even integer and 𝑎0, 𝑎=𝑎.
  • When 𝑛 is an even integer and 𝑎<0, 𝑎.isundenedover
  • When 𝑛 is an even integer and 𝑎, 𝑎=|𝑎|.

In our first two examples, we will demonstrate how to apply a combination of these properties to simplify an expression involving a root.

Example 1: Simplifying Algebraic Expressions Involving Exponents and Cube Roots

Simplify 64𝑚.

Answer

Recall that, for positive real numbers 𝑎 and 𝑏 and positive integers 𝑛, 𝑎𝑏=𝑎𝑏.

By applying this property in reverse, we can rewrite 64𝑚 as 64𝑚=64𝑚.

Next, we know that, for odd integers 𝑛, 𝑎=𝑎=𝑎.

Hence, 𝑚=𝑚.

Similarly, since 64=4, 64=4=4.

Combining these expressions, we obtain 64𝑚=4𝑚.

Example 2: Simplifying Algebraic Expressions Involving Exponents and Square Roots

Simplify 100𝑥.

Answer

When a root of the form 𝑥 is given with 𝑛 omitted, we assume 𝑛=2. This means we can simplify 100𝑥 by applying the property of 𝑛th roots with 𝑛=2, 𝑎𝑏=𝑎𝑏.

Thus, 100𝑥=100𝑥=10𝑥.

Finally, by writing 𝑥 as 𝑥 and using the property 𝑥=𝑥 for even 𝑛, we can simplify 𝑥 as follows: 𝑥=(𝑥)=𝑥.

Hence, 100𝑥=10𝑥.

Great care must be taken when finding powers of roots, such as 𝑥 or 𝑥. In the previous example, we demonstrated that (𝑥)=𝑥. Since we had an even power inside an even root, the operation was defined for all real values of 𝑥. If, however, we were simplifying 𝑥, we would need to perform the following steps: 𝑥=(𝑥)=||𝑥||.

In this case, 𝑥 inside the square root acts first, ensuring the radicand is positive for all values of 𝑥. This means we are not taking the even root of a negative number and, in turn, means the resulting function can only output positive values. Since 𝑥 is negative for values of 𝑥<0, we must include the absolute value when simplifying, as shown.

In the last two examples, we used a property of roots to express an 𝑛th root as the product of two unique 𝑛th roots. It is important to realize that we can extend this property to express a root as the product of three or more 𝑛th roots in order to simplify an expression. In other words, for positive real numbers 𝑎, 𝑏, and 𝑐 and positive integers 𝑛, 𝑎𝑏𝑐=𝑎𝑏𝑐.

Example 3: Simplifying Algebraic Expressions with More Than One Variable Involving Exponents and Square Roots

Write 25𝑎𝑏 in its simplest form.

Answer

Recall that, for positive real numbers 𝑎, 𝑏, and 𝑐 and positive integers 𝑛, 𝑎𝑏𝑐=𝑎𝑏𝑐.

Since 25𝑎𝑏 is equivalent to 25𝑎𝑏, we can rewrite it as shown: 25𝑎𝑏=25𝑎𝑏=5𝑎𝑏.

Next, by using the property 𝑥=|𝑥| for even 𝑛, we can write 𝑎=|𝑎|.

Similarly, 𝑏 can be written as 𝑏, so 𝑏=(𝑏)=||𝑏||.

This means 25𝑎𝑏=5|𝑎|||𝑏||. Since 5>0, 5=|5|, and the product of absolute value is the absolute value of the products, 25𝑎𝑏=||5𝑎𝑏||.

In the previous examples, we demonstrated how to use properties of roots to simplify expressions. Let’s now consider an equation involving exponents, 𝑦=16.

A solution to this equation is found by taking the square root, such that 𝑦=16=4.

However, if we substitute 𝑦=4 into the expression 𝑦, we obtain (4)=16. This means that 𝑦=4 is also a solution to the equation 𝑦=16. Therefore, when solving an equation 𝑦=𝑥, the solutions include both the positive and negative square roots of 𝑥. This gives rise to a subtle difference between the two seemingly equivalent statements for integer 𝑛, 𝑦=𝑥𝑦=𝑥.and

We can generalize this concept for 𝑛th roots where 𝑛 is an even number.

Theorem: Even and Odd 𝑛th Roots

Consider the equation 𝑦=𝑥 for real numbers 𝑦 and 𝑥 and positive integers 𝑛. The following solutions hold:

𝑛 Even𝑛 Odd
𝑥<0There are no real solutions to the equation.There is one solution,
𝑦=𝑥.
𝑥>0The solutions to the equation are
𝑦=±𝑥.

We have seen that we interpret the statements 𝑦=𝑥 and 𝑦=𝑥 differently. This gives rise to an additional definition, that of the principal 𝑛th root. This allows us to consider an 𝑛th root as a function by making it, by definition, a one-to-one mapping.

Definition: Principal 𝑛th Root

Every positive real number has a single positive 𝑛th root, defined 𝑥. This is known as the principal 𝑛th root.

Let’s demonstrate an application of the properties of even and odd 𝑛th roots in the next example.

Example 4: Simplifying and Solving Equations Involving 𝑛th Roots

Find the value (or values) of 𝑥 if 12𝑥=384.

Answer

To solve this equation, we will apply a series of inverse operations. First, we divide both sides by 12: 12𝑥12=38412𝑥=32.

Next, we recall that, given an equation 𝑥=𝑦 for real numbers 𝑥 and 𝑦 and positive integers 𝑛,

  • if 𝑛 is even, the solutions to the equation are 𝑥=±𝑦,
  • if 𝑛 is odd, there is just one solution, 𝑥=𝑦.

Since 𝑛=5 is odd, there is one solution to this equation, given by 𝑥=32=2.

Hence, 𝑥=2.

By combining the properties of 𝑛th roots and this theorem, we are able solve more complicated equations involving exponents. Let’s demonstrate this with one final example.

Example 5: Simplifying and Solving Equations Involving 𝑛th Roots

Find the value (or values) of 𝑥 given that 𝑥+95=144×3.

Answer

We will begin by evaluating the right-hand side of this equation. Since 144 and 3 are both square numbers, we can easily find the principal square root of their product, which is 144×3=1443=12×3=36.

Next, we recall that, given an equation 𝑥=𝑎 for real numbers 𝑥 and 𝑎 and positive integers 𝑛, if 𝑛 is even, the solutions to the equation are 𝑥=±𝑎. Since the expression on the left-hand side of the equation has an even exponent (𝑛=2), we will need to take the positive and negative square root of 36.

In particular, 𝑥+95=144×3𝑥+95=36𝑥+95=±36𝑥+95=±6.

To find the solutions to these two equations, we will next multiply both sides by 5 and then subtract 9: 𝑥+95=6𝑥+95=6𝑥+9=30𝑥+9=30𝑥=21,𝑥=39.

Hence, the solutions are 𝑥=21 and 𝑥=39.

Let’s finish by recapping some key concepts from this explainer.

Key Points

  • The 𝑛th root (or radical) of a number 𝑥, where 𝑛 is a positive integer, is a number that, when raised to the 𝑛th power, gives 𝑥. We can define this number as 𝑦, such that 𝑥=𝑦.
  • The 𝑛th root is given by 𝑦=𝑥.
  • Every positive real number has a single positive 𝑛th root, defined 𝑥. This is known as the principal 𝑛th root.
  • When 𝑎 and 𝑏 are well defined and are real numbers, 𝑎𝑏 is also defined such that 𝑎𝑏=𝑎𝑏.
  • If 𝑏0, then it is also the case that 𝑎𝑏=𝑎𝑏.
  • When 𝑛 is an odd integer, 𝑎=𝑎=𝑎.
  • When 𝑛 is an even integer and 𝑎0, 𝑎=𝑎.
  • When 𝑛 is an even integer and 𝑎<0, 𝑎.isundenedover
  • When 𝑛 is an even integer and 𝑎, 𝑎=|𝑎|.
  • The solutions to the equation 𝑦=𝑥 for real numbers 𝑥 and 𝑦 and positive integers 𝑛 are as follows:
    𝑛 Even𝑛 Odd
    𝑥<0There are no real solutions to the equation.There is one solution,
    𝑦=𝑥.
    𝑥>0The solutions to the equation are 𝑦=±𝑥.

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