10. Exponents and Powers

Do you know? Mass of earth is 5,970,000,000,000, 000, 000, 000, 000 kg. We have already learnt in earlier class how to write such large numbers more conveniently using exponents, as, 5.97 × 10²⁴ kg. We read 10²⁴ as 10 raised to the power 24. We know 2⁵ = 2 × 2 × 2 × 2 × 2. Let us now find what is 2⁻² equal to?

Exponents allow us to express large numbers compactly. For example, the mass of the Earth can be written as 5.97 x 10²⁴ kilograms instead of writing out all the zeros. The number 10²⁴ means that we take 10 and multiply it by itself 24 times. The example of calculating 2⁵ illustrates how exponents work with multiplication of the same base.

Imagine counting stars in the universe. Instead of saying there are a trillion stars, you can say there are about 1 x 10¹² stars, which is much simpler. This is similar to how we simplify the mass of the Earth with exponents.

Powers with Negative Exponents Exponent is a negative integer. You know that, 10² = 10 × 10 = 100. As the exponent decreases by 1, the value becomes one-tenth of the previous value. Continuing the above pattern we get, 10⁻¹ = 1/10 = 0.1 and similarly for 10⁻² = 1/(10 × 10) = 0.01.

Negative exponents represent reciprocals of positive exponents. For instance, 10⁻¹ is the same as 1/10, which equals 0.1. The negative exponent indicates that we are dealing with fractions. Similarly, 10⁻² becomes 1/100 or 0.01. Thus, negative exponents help in representing very small numbers easily.

Suppose you have a pizza that's cut into 10 slices. If you take one slice away, you have 1 out of 10, which can be represented as 10⁻¹. If you eat two slices, you're left with 1/100 of the pizza, which is represented as 10⁻².

In general, we can say that for any non-zero integer a, a⁻m = 1/a^m, where m is a positive integer. a⁻m is the multiplicative inverse of a^m.

This rule states that any base raised to a negative exponent is equal to the reciprocal of that base raised to the corresponding positive exponent. For example, if you have 2⁻³, it means 1/(2³) which is 1/8 or 0.125.

Think of it like owing someone money. If you owe $5, that is a negative amount in your bank account. Once you pay it back, you have the inverse - a positive balance. Similarly, negative exponents indicate the inverse relationship.

We learnt how to write numbers like 1425 in expanded form using exponents as 1 × 10³ + 4 × 10² + 2 × 10¹ + 5 × 10⁰. Let us see how to express 1425.36 in expanded form in a similar way.

When expressing numbers in expanded form using exponents, we break the number down by place value. For 1425.36, we can write it as 1 × 10³ for the thousand's place, 4 × 10² for the hundred's place, and so on. This helps to visualize each digit's contribution to the entire number.

Consider a bookshelf with 10 shelves. If each shelf can hold 10 books, you can express how many books you have by saying you have 1,000 on the top shelf, 400 on the next, and so forth. Naming the contribution of each shelf corresponds to expressing a number in its expanded form.

We have learnt that for any non-zero integer a, am × an = a(m+n), where m and n are natural numbers. Does this law also hold if the exponents are negative? Let us explore.

The laws of exponents show how to manipulate expressions with exponents. For instance, multiplying two powers with the same base results in adding their exponents. This law holds true even for negative exponents. For example, 3⁻² × 3⁻³ equals 3⁻(2+3) or 3⁻⁵.

Think about combining two boxes of different sizes. If one has a volume represented by a³ and the other by a², when combined, the total capacity can be described as a⁵. Just like combining various counts results in a total, combining exponents follows the same addition rule.

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