Class 9 Exam > Class 9 Notes > RD Sharma Solutions for Class 9 Mathematics > RD Sharma Solutions Ex-2.2, (Part - 1), Exponents Of Real Numbers, Class 9, Maths
RD Sharma Solutions Ex-2.2, (Part - 1), Exponents Of Real Numbers, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics PDF Download
1. Assuming that x,y,z are positive real numbers, simplify each of the following
2. Simplify
3. Prove that
4. Show that
Left hand side (LHS) = Right hand side (RHS)
Considering LHS,
Therefore, LHS = RHS
Hence proved
(ii) 
Left hand side (LHS) = Right hand side (RHS)
Considering LHS,
1
Therefore, LHS = RHS
Hence proved
5. If 2x = 3y = 12z , 
6. If 2x = 3y = 6−z , 
[by equating exponents]
7. If ax = by = cz and b2 = ac , then show that 
8. If 3x = 5y = (75)z, show that
We have,
3x = 2 − x [On equating exponents]
3x + x = 2
4x = 2
x = 2/4
x = 12
Here the value of x is 1/2
10. Find the values of x in each of the following
We have
= 4x = 4 [On equatinge xponent]
x = 1
Hence the value of x is 1
(ii). (23)4 = (22)x
We have
(23)4 = (22)x
= 23×4 = 22×x
12 = 2x
2x = 12 [On equating exponents]
x = 6
Hence the value of x is 6
We have
x = 3 [on equating exponents]
Hence the value of x is 3
(iv) 5x−2×32x−3 = 135
We have,
⇒ x − 2 = 1,2x − 3 = 3 [On equating exponents]
⇒ x = 2 + 1,2x = 3 + 3
⇒ x = 3, 2x = 6 ⇒ x = 3
Hence the value of x is 3
(v). 2x−7×5x−4 = 1250
We have
2x−7×5x−4 = 1250
⇒ 2x−7×5x−4 = 2×625
⇒ 2x−7×5x−4 = 2×54
⇒ x−7 = 1 ⇒ x = 8,x−4 = 4 ⇒ x = 8
Hence the value of x is 8
4x+1 = −15
4x = −15−1
4x = −16
x =−16/4
x = −4
Hence the value of x is 4
(vii). 52x+3 = 1
52x+3 = 1×50
2x+3 = 0 [By equatinge xponents]
2x = −3
x = −3/2
Hence the value of x is −3/2
[By equating exponents]
Hence the value of x is 4
Hence the value of x is 7
The document RD Sharma Solutions Ex-2.2, (Part - 1), Exponents Of Real Numbers, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics is a part of the Class 9 Course RD Sharma Solutions for Class 9 Mathematics.
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FAQs on RD Sharma Solutions Ex-2.2, (Part - 1), Exponents Of Real Numbers, Class 9, Maths - RD Sharma Solutions for Class 9 Mathematics
| 1. What are exponents of real numbers? |  |
Ans. Exponents of real numbers are a way to represent repeated multiplication of a number by itself. An exponent tells us how many times the base number is being multiplied by itself. For example, in the expression 2^3, the base number is 2 and the exponent is 3. It means that we multiply 2 by itself 3 times, resulting in 2^3 = 2 x 2 x 2 = 8.
| 2. How do you simplify expressions with exponents? | |
Ans. To simplify expressions with exponents, we use the properties of exponents. Here are some important properties:
- Product Rule: When multiplying two numbers with the same base, we add their exponents. For example, a^m * a^n = a^(m+n).
- Quotient Rule: When dividing two numbers with the same base, we subtract their exponents. For example, a^m / a^n = a^(m-n).
- Power Rule: When raising a number with an exponent to another exponent, we multiply the exponents. For example, (a^m)^n = a^(m*n).
- Zero Rule: Any number (except zero) raised to the power of zero is equal to 1. For example, a^0 = 1.
| 3. What is the difference between a positive exponent and a negative exponent? | |
Ans. A positive exponent indicates that the base number is being multiplied by itself a certain number of times. For example, 2^3 means multiplying 2 by itself 3 times, which is equal to 2 x 2 x 2 = 8.
On the other hand, a negative exponent indicates that the base number is being divided by itself a certain number of times. For example, 2^-3 means dividing 1 by 2 three times, which is equal to 1 / (2 x 2 x 2) = 1/8.
So, while a positive exponent leads to a larger value, a negative exponent leads to a smaller value.
| 4. How do you simplify expressions with negative exponents? | |
Ans. To simplify expressions with negative exponents, we can use the reciprocal property of exponents. When a number with a negative exponent is in the numerator, we can move it to the denominator and change the sign of the exponent to positive. Similarly, when a number with a negative exponent is in the denominator, we can move it to the numerator and change the sign of the exponent to positive.
For example, if we have 3^(-2), we can rewrite it as 1 / (3^2) = 1/9. And if we have 1 / 2^(-3), we can rewrite it as 2^3 = 8.
| 5. Can exponents of real numbers be fractions or decimals? | |
Ans. Yes, exponents of real numbers can be fractions or decimals. The rules of exponents apply to any real number, not just whole numbers. For example, 2^(1/2) is the square root of 2, which is approximately 1.414. Similarly, 2^0.5 is also the square root of 2.
Fractional or decimal exponents represent roots or powers that are not whole numbers. They can be used to calculate values like cube roots, fourth roots, and so on. The properties of exponents, such as the product rule and quotient rule, still apply to these fractional or decimal exponents.
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