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Powers and Roots

Powers and Roots

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Powers and Roots

In mathematics, powers are used as a way to multiply a number by itself multiple times. Simply put, a power is the exponent that a variable or number is raised to. For instance, the expression x² is read as "x to the power of 2" or "x squared," indicating that x is multiplied by itself twice, or as many times as the value of the exponent.

Let's take the value of x as 5. In this case, we can compute x² by multiplying 5 by itself:

  • x² = 5 x 5 = 25

In a similar manner, we can compute x³ by multiplying 5 by itself three times:

  • x³ = 5 x 5 x 5 = 125

It's important to note that if a variable has no exponent, it is considered to be 1. For instance, x¹ is equal to x. Additionally, any variable raised to the power of 0 equals 1. For a more comprehensive explanation of the rules surrounding exponents, consult the Exponential Rules.

Understanding Roots

Roots are essentially the opposite of powers. In simpler terms, they are the inverse of powers. To find the root of a number n, we must determine a number that when multiplied by itself n times, yields the number inside the radical symbol (x).

1. Square Root

The square root is the most commonly used root, represented by the radical symbol (√). To find the square root of a number, we must determine a number that, when multiplied by itself, gives us the number inside the symbol.

For example, to find the square root of 25, we must determine a number that when multiplied by itself is equivalent to 25. We know that 5 x 5 = 25 or (-5) x (-5) = 25. Therefore, the square root of 25 can be either 5 or -5. This is why the result is expressed as ±5. But why is it ±5? This is because the square root of a positive number can have both positive and negative solutions. However, the square root of a negative number does not have a real solution and requires the use of imaginary numbers.

The square root of perfect squares always results in an integer. It's helpful to remember the first ten perfect squares when working with expressions containing powers and roots. They are as follows:

  • √1 = 1
  • √4 = 2
  • √9 = 3
  • √16 = 4
  • √25 = 5
  • √36 = 6
  • √49 = 7
  • √64 = 8
  • √81 = 9
  • √100 = 10

If the number inside the square root has a square number as a factor, it can be simplified. For instance, √20 can be simplified to 2√5 because 20 = 4 x 5. To learn more about simplifying square roots, read about Surds.

2. Cube Root

The cube root is the inverse of cubing a number. As the name suggests, it is represented by the ³√ symbol. This means finding a number that, when multiplied by itself three times, yields the number inside the symbol.

For example, to find the cube root of 8, we must determine a number that when multiplied by itself three times is equal to 8. In this case, the only possible answer is 2 since (-2) x (-2) x (-2) = -8.

Cube roots can be taken for both positive and negative numbers, unlike square roots.

3. Other Roots

There are also other types of roots, such as fourth roots, fifth roots, and so on. Writing powers as roots and roots as powers requires an understanding of fractional exponents.

Fractional Exponents

Fractional exponents are equivalent to roots. They follow the following exponential rule:

xm/n = n√xm

This expression can be used to rewrite any root as a fractional exponent.

Evaluating and Simplifying Expressions with Powers and Roots

Now that we have a grasp of fractional exponents and the rules for handling exponents, we can evaluate and simplify expressions involving powers and roots. Here are some examples:

Example 1

Evaluate or simplify: √16 x (√9 + 1)

By recalling perfect squares, we can substitute √16 for 4. Additionally, √9 can be substituted for 3.

Understanding Powers and Roots

Powers and Roots are mathematical concepts that involve raising a number or variable to an exponent and finding the inverse operation. In simple terms, powers involve repeated multiplication of a number by itself, while roots involve finding the number that was multiplied by itself to get a given result.

One way to convert powers into roots is by using the fractional exponent rule, which states that the bth root of x to the power of a is equal to x to the power of a over b. This rule enables us to transform powers into roots and vice versa.

To calculate a power, we simply multiply the number or variable by itself as many times as the exponent. For instance, a number raised to the power of 3 is equivalent to that number multiplied by itself three times.

Simplifying roots and powers can be made easier by familiarizing ourselves with some key concepts. For example, memorizing the square root of perfect squares can aid in simplifying square roots. Additionally, having a basic understanding of the exponential rules can also help in simplifying roots and powers.

Example 2

Evaluate or Simplify: 1252/3

We can rewrite the root as a fractional exponent using the exponential rule:

  • 1252/3 = (1251/3)2
  • Using the rules of exponents, we get:
  • (1251/3)3 = 53 = 125

Example 3

Evaluate or Simplify: 82/3 + 81/3

Utilizing the exponential rule, we get:

  • 82/3 + 81/3 = (81/3)2 + 81/3
  • Let's simplify by replacing 81/3 with x.
  • (81/3)2 + 81/3 = x2 + x = 9
  • Solving the quadratic equation x2 + x - 9 = 0, we get x = 2 or -3.
  • Substituting back x for 81/3, we arrive at the two solutions: 81/3 = 2 or -3.

Key Takeaways

Here are some key takeaways from this article:

  • A power is the exponent that a number or variable is raised to.
  • A root is the inverse of a power.
  • Odd roots have one solution, while even roots have two solutions.
  • Square roots can only be taken with positive numbers without the use of imaginary numbers.
  • Cube roots can be taken for both positive and negative numbers.
  • Knowing the square roots of perfect squares and the exponential rules can be beneficial in evaluating or simplifying expressions with powers and roots.

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