Exponents | Mathematics for ACT PDF Download

An exponent of a number shows how many times we are multiplying a number by itself. For example, 3⁴ means we are multiplying 3 four times. Its expanded form is 3 × 3 × 3 × 3. Exponent is also known as the power of a number. It can be a whole number, fraction, negative number, or decimals.

What are Exponents?

The exponent of a number shows how many times the number is multiplied by itself. For example, 2 × 2 × 2 × 2 can be written as 2⁴, as 2 is multiplied by itself 4 times. Here, 2 is called the "base" and 4 is called the "exponent" or "power." In general, xn means that x is multiplied by itself for n times.

Here, in the term xⁿ,

• x is called the "base"
• n is called the "exponent"
• xⁿ is read as "x to the power of n" (or) "x raised to n".

Some examples of exponents are as follows:

• 3 × 3 × 3 × 3 × 3 = 3⁵
• -2 × -2 × -2 = (-2)³
• a × a × a × a × a × a = a⁶

Exponents are important because, without them, when a number is repeated by itself many times it is very difficult to write the product. For example, it is very easy to write 5⁷ instead of writing 5 × 5 × 5 × 5 × 5 × 5 × 5.

Properties of Exponents

The properties of exponents or laws of exponents are used to solve problems involving exponents. These properties are also considered as major exponents rules to be followed while solving exponents. The properties of exponents are mentioned below.

• Law of Product: a^m× aⁿ = a^m+n
• Law of Quotient: a^m/aⁿ = a^m-n
• Law of Zero Exponent: a⁰ = 1
• Law of Negative Exponent: a^-m = 1/a^m
• Law of Power of a Power: (a^m)ⁿ = a^mn
• Law of Power of a Product: (ab)^m = a^mb^m
• Law of Power of a Quotient: (a/b)^m = a^m/b^m

Negative Exponents

A negative exponent tells us how many times we have to multiply the reciprocal of the base. For example, if it is given that a^-n, it can be expanded as 1/aⁿ. It means we have to multiply the reciprocal of a, i.e 1/a 'n' times. Negative exponents are used while writing fractions with exponents. Some of the examples of negative exponents are 2 × 3^-9, 7^-3, 67^-5, etc. We can convert these into positive exponents as follows:

• 2 × 3^-9 = 2 × (1/3⁹) = 2 / 3⁹
• 7^-3 = 1/7³
• 67^-5 = 1/67⁵

Exponents with Fractions

If an exponent of a number is a fraction, it is known as a fractional exponent. Square roots, cube roots, n^th root are parts of fractional exponents. Number with power 1/2 is termed as the square root of the base. Similarly, a number with a power of 1/3 is called the cube root of the base. Some examples of exponents with fractions are 5^2/3, -8^1/3, 10^5/6, etc. We can write these as:

• 5^2/3 = (5²)^1/3 = 25^1/3 = ∛25
• -8^1/3 = ((-2)³)^1/3 = -2
• 10^5/6 = (10⁵)⁶ = ⁶√10⁵ = ⁶√100000

Decimal Exponents

If an exponent of a number is given in the decimal form, it is known as a decimal exponent. It is slightly difficult to evaluate the correct answer of any decimal exponent so we find the approximate answer for such cases. Decimal exponents can be solved by first converting the decimal in fraction form. For example, 4^1.5 can be written as 4^3/2 which can be simplified further to get the final answer 8. i.e., 4^3/2 = (2²)^3/2 = 2³ = 8.

Scientific Notation with Exponents

Scientific notation is the standard form of writing very large numbers or very small numbers. In this, numbers are written with the help of decimal and powers of 10. A number is said to be written in scientific notation when a number between 0 to 10 is multiplied by a power of 10. In the case of a number greater than 1, the power of 10 will be a positive exponent, while in the case of numbers less than 1, the power of 10 will be negative. Let's understand the steps for writing numbers in scientific notation with exponents:

• Step 1: Put a decimal point after the first digit of the number from the left. If there is only one digit in a number excluding zeros, then we don't need to put decimal.
• Step 2: Multiply that number with a power of 10 such that the power will be equal to the number of times we shift the decimal point.

By following these two simple steps we can write any number in standard form with exponents, for example, 560000 = 5.6 × 10⁵, 0.00736567 = 7.36567 × 10^-3.

Example: The dimensions of a wardrobe are given in terms of exponents such as x⁵ units, y³ units, and x⁸ units. Find its volume.
Solution: The given dimensions of the wardrobe are in form of exponents, i.e., length = x⁵ units, width = y³ units, and height = x⁸ units.
The volume of the wardrobe is, volume= lwh.
So, by substituting the values, the volume of the wardrobe is x⁵ × y³ × x⁸ = x¹³y³ (using exponents formula = a^m × aⁿ = a^(m+n)).
Therefore, the volume of the wardrobe is x¹³y³ cubic units.