Powers - Maths

Powers are “numbers raised to an exponent, indicating how many times the base number is multiplied by itself.” 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128, which can also be expressed as 2^7, illustrating how powers simplify repeated multiplication. Understanding powers is essential in algebra, as they are used to represent large numbers and to solve equations more efficiently.

The powers of ten are really easy - the power tells you the number of zeros.

10^1 = 10, representing a single zero and the base number itself.

10² = 100, which shows that there are two zeros following the base number, 1.

10³ = 1000, indicating that there are three zeros after the base number, 1.

10^6 = 1000000, which signifies that there are six zeros following the base number, 1.

Use the exponent as a shorthand to represent very large or very small numbers efficiently, simplifying calculations and comparisons.

Anything to the power 1 is just itself, for example, 10¹ = 10, and this property holds true for all numbers regardless of whether they are whole numbers, fractions, or decimals.

1 to any power is still 1, and thus, 1² = 1, 1³ = 1, and so on.

Anything to the power 0 is just 1, so for any non-zero number, such as 5⁰ = 1 and (-3)⁰ = 1, this rule applies universally and is a fundamental property of exponents.

When Multiplying, you add the powers. For example, when calculating 2² × 2³, you would add the exponents to get 2^{2+3} = 2⁵ = 32.

When Dividing, you subtract the powers. For instance, in the case of 5³ ÷ 5², you would subtract the exponents, leading to 5^{3-2} = 5¹ = 5.

When Raising one power to another, you multiply the powers. For example, (3²)³ = 3^{2*3} = 3^6 = 729.

Fractions - Apply the power to both top and bottom. e.g. ( \left( \frac{2}{5} \right)^{3} = \frac{2^{3}}{5^{3}} = \frac{8}{125} ) which shows how to handle powers with fractions.

EXAMPLE
a = 5^9 and b = 5^4 × 5². What is the value of a/b?

To simplify this expression, we first need to rewrite b using the properties of exponents: b = 5^{4 + 2} = 5^{6}. Hence, the expression becomes a/b = 5^{9}/5^{6}, and by applying the quotient of powers rule, we get a/b = 5^{9-6} = 5^{3}. Therefore, the final value is 5^3 = 125.

To find a negative power - turn it upside-down.

People have real difficulty understanding the concept of negative exponents since it implies that the base is in the denominator, such as 5^{-n} = 1/5^{n}.

E.g. 7² = 49 and 7^{-2} = 1/7^{2} = 1/49.

(3/5)^-2 = 1/(3/5)^{2} = 1/(9/25) = 25/9.

Practise these power rules - you never know when they might come in handy in future mathematical problems or real-life applications.

If you can add, subtract, multiply, and divide numbers with confidence, mastering powers will be a significant boost to your overall mathematical skills.

Q1) Find 3³ + 4² without a calculator. To solve this, calculate each power first: 3³ = 27 and 4² = 16. Then, add the results together: 27 + 16 = 43.

Use your calculator to find 6.2³.

To find 6.2³, you can multiply 6.2 by itself three times: 6.2 × 6.2 × 6.2 = 238.328.

Write ten thousand as a power of 10. 10,000 can be expressed as 10^4.

Simplify: a) 4² x 4³ = 4^{2+3} = 4^5.

b) 7^6/7³ = 7^{6-3} = 7^3.

c) (q²)^4 = q^8

Without using a calculator, find

6³ x 6^5/6^6 = 6² = 36

2^-4 = 1/16