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Last modified on October 17th, 2024

An exponent is a mathematical notation that represents how many times a number, called the base, is multiplied by itself.

For example, in 5 × 5 × 5, 5 is multiplied 3 times. So, in exponent form, it is written as 5^{3}, where 5 is the base.

The number written above 5 in superscript (i.e., 3) is called the exponent. Often called the index (plural: indices) or power, it represents how many times the number is repeated in multiplication, known as exponentiation.

The above expression 5^{3} is read as ‘5 to the power 3,’ or ‘5 to the 3rd,’ or ‘5 cubed.’

Here are a few more examples of exponents:

- 2
^{5}= 2 × 2 × 2 × 2 × 2 = 32 - 3
^{2}= 3 × 3 = 9 - 6
^{3}= 6 × 6 × 6 = 216

In general, if *x* is an integer raised to a positive integer *n*, then it is expressed as x^{n}

The symbol ^ is also used to represent exponents while typing. This is called the ‘carrot.’

**Find the value of 10 ^{5} **

Solution:

Here,

10^{5} = 10 × 10 × 10 × 10 × 10 = 100000

Some important laws that are used while solving expressions involving exponents are listed:

Properties | Mathematical Form |

Product of Powers | x^{m} × x^{n} = x^{m + n} |

Quotient of Powers | ${\dfrac{x^{m}}{x^{n}}=x^{m-n}}$ |

Power of a Power | (x^{m})^{n} = x^{mn} |

Power of a Product | (xy)^{m} = x^{m}y^{m} |

Power of a Quotient | ${\dfrac{x^{m}}{y^{m}}=\left( \dfrac{x}{y}\right) ^{m}}$ |

Zero Exponent | x^{0} = 1 |

One Exponent | x^{1} = x |

Negative Exponent | ${x^{-m}=\left( \dfrac{1}{x}\right) ^{m}}$ |

Fractional Exponent | ${x^{\dfrac{m}{n}}}$ = ${\left( \sqrt[n]{x}\right) ^{m}}$ or ${\sqrt[n]{x^{m}}}$ |

These rules make it easier to perform operations with exponents without needing to manually multiply the base repeatedly.

**Find the product of 2 ^{3} × 2^{4}**

Solution:

Given 2^{3} × 2^{4}

As we know, x^{m} × x^{n} = x^{m + n}

Here, 2^{3} × 2^{4} = 2^{3 + 4} = 2^{7} = 128

**Solve the expression: ${\dfrac{5^{6}}{5^{2}}}$**

Solution:

Given ${\dfrac{5^{6}}{5^{2}}}$

As we know, ${\dfrac{x^{m}}{x^{n}}=x^{m-n}}$

Here, ${\dfrac{5^{6}}{5^{2}}}$ = ${5^{6-2}}$ = 5^{4} = 625

When the exponent of a number is negative, it is called a negative exponent.

While a positive exponent tells us how many times to multiply a base by itself, a negative exponent tells us to take the reciprocal (or ‘flip’) of the base raised to its corresponding positive exponent. Thus, a negative exponent inverts the base, shifting it from the numerator to the denominator (or vice versa) of a fraction.

For example,

2^{-3} can be expressed as ${\dfrac{1}{2^{3}}}$

When the exponent of a number is in decimals, it is called a decimal exponent. It is written as a combination of integer powers and roots.

For example,

25^{0.5} is broken down as

= ${25^{\dfrac{1}{2}}}$, where 25^{0.5} represents the square root of the base 25 (i.e., ${\sqrt{25}}$)

= 5

Sometimes, evaluating decimal exponents can be challenging. In such cases, we find the approximate value.

For example,

5^{0.5} = ${5^{\dfrac{1}{2}}}$ ≈ 2.236

**Simplify: 64 ^{0.75}**

Solution:

Given 64^{0.75} = ${64^{\dfrac{3}{4}}}$

Now, applying the fractional exponent rule, ${x^{\dfrac{m}{n}}}$ = ${\sqrt[n]{x^{m}}}$

Here, ${64^{\dfrac{3}{4}}}$ = ${\sqrt[4]{64^{3}}}$ = ${\sqrt[4]{262144}}$ ≈ 22.63 (rounded to two decimal places)

**Problem – **Evaluating **EXPONENTS** with **BINARY EXPONENTIATION**

**Calculate 5 ^{15}**

Solution:

Here, 5^{15} = (5^{3})^{5}

As it is difficult to multiply 5^{3} for 5 times, here we will simplify the exponent 15 in binary form.

15 = 1 × 2^{3} + 1 × 2^{2} + 1 × 2^{1} + 1 × 2^{0} = 8 + 4 + 2 + 1, which simplifies the expression to:

5^{15} = 5^{(8 + 4 + 2 + 1)} = 5^{8} × 5^{4} × 5^{2} × 5^{1}

Now, 5^{8} = (5^{4})^{2} = 625^{2} = 625 × 625

Thus, 5^{10} can also be expressed as 625 × 625 × 625 × 25 × 5

**Evaluate 2 ^{10}**

Solution:

Given 2^{10}

Converting the exponent 10 into its binary form, we get

10 = 1 × 2^{3} + 0 × 2^{2} + 1 × 2^{1} + 0 × 2^{0} = 8 + 2

Here, 2^{10} = 2^{8 + 2} = 2^{8} × 2^{2} = 256 × 4 = 1024

Thus, 2^{10} = 1024

Exponents are often used in scientific notation to represent very large or very small numbers using powers of 10. In this system, a positive exponent is used for numbers greater than 1, while a negative exponent is used for numbers less than 1.

To express a number in scientific notation, we follow the given steps:

**Placing the decimal point after the first non-zero digit**in the number.**Multiplying by 10 raised to the power**that represents how many places the decimal point is shifted

For large numbers, the exponent is positive, indicating how many times the decimal point is shifted to the left. For small numbers, the exponent is negative, showing the number of times the decimal point is shifted to the right.

Let us express 7600000000000 and 0.00000003874 in scientific notation form.

7600000000000 = 7.6 × 10^{12}

0.00000003874 = 3.874 × 10^{-8}

To have a better understanding of the use of exponents in writing numbers in scientific notation, click here**.**

**Write the given numbers in scientific notation:****a) 980000 ****b) 0.0000543****c) 1210000**

Solution:

a) 980000 = 9.8 × 10^{5}

b) 0.000000543 = 5.43 × 10^{-7}c)1210000 = 1.21 × 10^{6}

Last modified on October 17th, 2024