Powers with Negative Exponents

You have learnt in your earlier classes that am means:

• ‘a’ to the power ‘m’
• ‘ a ‘ multiplied by itself ‘m’ times.

Here ‘a’ is called the base and ‘m’ is called the exponent. am is called the exponential form.

Remember:

• Odd powers of negative numbers give a negative result.
• Even powers of negative numbers give a positive result.

Example,

(-2)³ = -8, which is negative

(-2)⁴ = 16, which is positive

You know how to prime factorise a number. We use the method of prime factorisation to write the given number as a product of its prime factors in exponential form.

Consider the number 432. Let us prime factorise 432.

432 = 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2⁴ × 3³

Now, observe the following:

Let us continue the pattern above.

So,

x^-m = 1 x^m

where x is any non-zero integer and m is a positive integer.

We can also say that,

• x^-m is the reciprocal or multiplicative inverse of x^m
• x^m is the reciprocal or multiplicative inverse of x^-m

For example,

In science, we deal with very large and very small numbers. For example,

• One gram of hydrogen contains 602,000,000,000,000,000,000,000 atoms.
• Radius of a hydrogen atom is 0.000000000025 metre.

We use positive exponents to write very large numbers. In the same manner, we use negative exponents to write very small numbers.

You know how to evaluate powers with positive exponents. For example, to find (-2)⁵ we multiply (-2) by itself 5 times.

Let us evaluate 2^-5

Step 1 : Convert the negative exponent to positive exponent.

We know that x^-m = 1 x^m

So, 2^-5 = 1 2⁵

Step 2 : Evaluate the expression by repeated multiplication.

Hence, 2^-5 = 1 32

Now, consider the decimal number 123.456. Its expanded form is:

123.456 = 1 × 100 + 2 × 10 + 3 × 1 + 4 × 1 10 + 5 × 1 100 + 6 × 1 1000

= 1 × 10² + 2 × 10¹ + 3 × 10⁰ + 4 × 1 10¹ + 5 × 1 10² + 6 × 1 10³

Since x-m = 1 x^m

Laws of Exponents

Laws of exponents hold good even if exponents are negative.

Now, let us simplify (6⁴ ÷ 6^-5)3 × 6^-3

Let us observe the expression and identify the laws that are to be used to simplify it.

Let us apply these rules.

(6⁴ ÷ 6^-5)³ × 6^-3

= (6^{4 – (-5)})³ × 6^-3

= 6^{9 × 3} × 6 ^-3

= 6²⁷ × 6^-3

= 6^{27 + (-3)}

= 6²⁴

Now consider this equation.

We can see that the LHS and RHS are exponential numbers raised to the same base. So, we can say that y = 5.

When two exponential numbers have the same base, then their exponents are equal.

Expressing Very Small and Very Large Numbers in the Standard Form

You can express very large numbers in the standard form. For example:

It is difficult to work with large numbers. Expressing large numbers in standard form simplifies the operations performed on them like comparing, adding, subtracting, etc.

Can we write small numbers in the standard form? We use negative exponents to write very small numbers in the standard form. For example,

Pollen grains cause allergies in many people, but they usually are too small to see. A pollen grain is approximately, only 0.00002 meters in diameter. How is this number written in the standard notation?

0.00002 = 2 100000 = 2 10⁵ = 2 × 10^-5

The diameter of a pollen grain written in the standard form is 2 × 10^-5m.

Let us now understand how to write the number in the standard form back to its usual form.

The size of a plant cell is 1.275 × 10^-5 metres. Let us write in the numerical (usual) form.

Hence, the size of a plant cell is 0.00001275 metre.

You know how to compare two numbers. Let us now understand how to compare numbers written in the standard form.

Compare 7.5 × 10⁵ and 2.5 × 10⁷

The number with the smaller exponent is smaller. Hence, 7.5 × 10⁵ < 2.5 × 10⁷

Compare 7.5 × 10^-5 and 2.5 × 10^-7

The number with the greater exponent is the greater.

Compare 7.5 × 10^-5 and 2.5 × 10^-5

Both the numbers have the same exponent, observe the decimal number part. We can see that 7.5 > 2.5