CBSE Class 7 Mathematics Exponents and Powers Notes

CBSE Class 7 Laws of Exponents Concepts. Learning the important concepts is very important for every student to get better marks in examinations. The concepts should be clear which will help in faster learning. The attached concepts made as per NCERT and CBSE pattern will help the student to understand the chapter and score better marks in the examinations.

Laws of Exponents

EXPONENTS AND POWERS

1.What is an Exponent?

An exponent is simply shorthand for multiplying that number of identical factors.

So 4³ is the same as (4)(4)(4).

 And x³ is just three factors of x, (x)(x)(x).

Caution: Remember the order of operations. Exponents are the first operation (in the absence of grouping symbols like parentheses), so the exponent applies only to what it’s directly attached to. 3x³ is 3(x)(x)(x), not (3x)(3x)(3x). If we wanted (3x)(3x)(3x), we’d need to use grouping: (3x)³.

Negative Exponents: 4^{– 3} is the same as 1/(4³), and x ^–3 = 1/x³.

As you know, you can’t divide by zero. So there’s a restriction that x⁻ⁿ = 1/xⁿ only when x is not zero. When x = 0, x⁻ⁿ is undefined.

Fractional Exponents: A fractional exponent — specifically, an exponent of the form 1/n — means to take the nth root instead of multiplying or dividing. For example, 4^(1/3) is the 3rd root (cube root) of 4(1/3)=

Arbitrary Exponents: You can’t use counting techniques on an expression like 6^0.1687 or 4.3^π. Instead, these expressions are evaluated using logarithms.

You Try It!

1. Write 11³ as a multiplication.
2. Write j^-7 as a fraction, using only positive exponents.
3. What’s the value of 100^½?
4. Evaluate −5^-2 and (−5)^-2.

2..     Multiplying and Dividing Powers

Two Powers of the Same Base: Suppose you have (x⁵)(x⁶); how do you simplify that? Just remember that you’re counting factors.

x⁵ = (x)(x)(x)(x)(x)  and  x⁶ = (x)(x)(x)(x)(x)(x)

 Now multiply them together:

(x⁵)(x⁶) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x¹¹

When the bases are the same, you find the new power by just adding the exponents:

Powers of Different Bases: Caution! The rule above works only when multiplying powers of the same base. For instance, 

(x³)(y⁴) = (x)(x)(x)(y)(y)(y)(y)

If you write out the powers, you see there’s no way you can combine them.

Except in one case:  If the bases are different but the exponents are the same, then you can combine them. Example:

(x³)(y³) = (x)(x)(x)(y)(y)(y)

But you know that it doesn’t matter what order you do your multiplications in or how you group them. Therefore,

(x)(x)(x)(y)(y)(y) = (x)(y)(x)(y)(x)(y) = (xy)(xy)(xy)

But from the very definition of powers, you know that’s the same as (xy)³. And it works for any common power of two different bases:

(x³) (y³) = (xy)³

Dividing Powers: What about dividing? Remember that dividing is just multiplying by 1-over-something. So all the laws of division are really just laws of multiplication. The extra definition of

x^-n as 1/xⁿ comes into play here.

Example: What is x⁸ ÷ x⁶?

Well, there are several ways to work it out. One way is to say that x ÷x⁶ = x⁸(1/x⁶), but using the definition of negative exponents that’s just x⁸(x^-6). Now use the product rule (two powers of the same base) to rewrite it as x^8+(-6), or x^8-6, or x². Another method is simply to go back to the definition: x⁸÷x⁶ = (xxxxxxxx)÷(xxxxxx) = (xx)(xxxxxx)÷(xxxxxx) = (xx)(xxxxxx÷xxxxxx) = (xx)(1) = x². However you slice it, you come to the same answer: for division with like bases you subtract exponents, just as for multiplication of like bases you add exponents:

 But there’s no need to memorize a special rule for division: you can always work it out from the other rules or by counting.

In the same way, dividing different bases can’t be simplified unless the exponents are equal. x³÷y² can’t be combined because it’s just xxx/yy;  But x³÷y³ is xxx/yyy, which is (x/y)(x/y)(x/y), which is (x/y)³.

Negative Powers on the Bottom: What about dividing by a negative power, like y⁵/x⁻⁴? Use the rule you already know for dividing:   

y⁵/x⁻⁴  = y⁵ × x⁴ 

You Try It!

Write each of these as a single positive power.

5. a⁷ ÷ b⁷
6. 11² × 2³
7. 8³ x³
8. 5⁴ × 5⁶
9. p¹¹ ÷ p⁶
10. r^-11 ÷ r^-2 

3. Powers of Powers

What do you do with an expression like (x⁵)⁴? There’s no need to guess — work it out by counting.

(x⁵)⁴ = (x^{5 x 4} )= x²⁰

When you have a power of a power, the combined exponent is formed by multiplying.

You Try It!: Perform the operations to remove parentheses:

11. (x⁴)^-5

12. (5x²)³

4. The Zero Exponent

You probably know that anything to the 0 power is 1. But now you can see why. Consider x⁰.

By the division rule, you know that x³/x³ = x⁽³⁻³⁾ = x⁰. But anything divided by itself is 1, so x³/x³ = 1. Things that are equal to the same thing are equal to each other: if x³/x³ is equal to both 1 and x⁰, then 1 must equal x⁰. Symbolically,

x⁰ = x⁽³⁻³⁾ = x³/x³ = 1

There’s one restriction. You saw that we had to create a fraction to figure out x⁰. But division by 0 is not allowed, so our evaluation works for anything to the 0 power except zero itself:

Evaluating 0⁰ is a separate advanced topic

You Try It!

What is the value of each of these?

13. (a⁶b⁸c¹⁰ / a⁵b⁶d⁷)⁰

14. 17x⁰

5. Radicals

The laws of radicals are traditionally taught separately from the laws of exponents, and frankly I’ve never understood why. A radical is simply a fractional exponent: the square (2nd) root of x is just x^1/2, the cube (3rd) root is just x^1/3, and so on.

Example: You know that the cube (3rd) root of x is x^1/3 and the square root of that is (x^1/3)^1/2. Then use the power-of-a-power rule to evaluate that as x(^1/3)(1/2) = x^(1/6) which is the 6th root of x.

Example: Because the square root is the 1/2 power, and the product rule for the same power of different bases tells you that (x^1/2)(y^1/2) = (xy)^1/2.

Fractional or Rational Exponents So far we’ve looked at fractional exponents only where the top number was 1. How do you interpret x2/3, for instance? Can you see how to use the power rule?

Since 2/3 = (2)(1/3), you can rewrite x^2/3 = x^(2)(1/3) = (x²)^1/3 It works the other way, too: 2/3 = (1/3)(2), so x2/3 = x^(1/3)(2) = (x^1/3)²

When a power and a root are involved, the top part of the fractional exponent is the power and the bottom part is the root.

You Try It!

15. Write √x⁵ as a single power.

16. Simplify ³√(a⁶b⁹) (That’s the cube root or third root of a⁶b⁹.)

17. Find the numerical value of 27^4/3 without using a calculator.

6. Laws of Exponents:

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