What is the value of (\frac{5^{-2}+5^{-3}}{5^{-4}})?
Correct answer: A
Here (5^{-2}+5^{-3}=\frac{1}{25}+\frac{1}{125}=\frac{6}{125}), and (5^{-4}=\frac{1}{625}). Division gives (30).
Mathematics
Operations on real numbers and the laws of exponents
वास्तविक संख्याओं पर संक्रियाएँ और घातांक के नियम
In this Class 10 Mathematics topic from the Polynomials chapter, students strengthen their understanding of operations on real numbers and the laws of exponents. They learn to add, subtract, multiply and divide numerical expressions accurately, use exponent rules for products, quotients and powers, and simplify expressions involving positive, zero and negative exponents where appropriate. The topic builds fluency in working with polynomial terms, comparing equivalent forms, and checking calculations through properties such as commutativity, associativity and distributivity. Examples connect numerical rules with algebraic manipulation and related exercises.
TOPIC PRACTICE
Quiz this set
Up to 16 questions from this page. Select your focus, then start.
16 questions
Choose questions
Expert · Level 6
View options
- (30)
- (25)
- (20)
- (15)
Expert · Level 6
View options
- 5
- 11
- 17
- 22
Expert · Level 6
View options
- \(9r^{6}s^{-8}\)
- \(\frac{1}{9}r^{-6}s^{8}\)
- \(9r^{-6}s^{8}\)
- \(\frac{1}{9}r^{6}s^{-8}\)
Expert · Level 6
View options
- (\frac{9}{2})
- (\frac{7}{2})
- (4)
- (5)
Expert · Level 6
View options
- \(a^p\cdot a^q=a^{p+q}\)
- \((a^p)^q=a^{p+q}\)
- \(a^p+a^q=a^{p+q}\)
- \(\dfrac{a^p}{a^q}=a^{p/q}\)
Expert · Level 6
View options
- (3)
- (6)
- (9)
- (12)
Expert · Level 6
View options
- (16\sqrt{17})
- (8\sqrt{17})
- (32\sqrt{17})
- (17)
Expert · Level 6
View options
- \(\frac{4x^{3}y^{4}}{5}\)
- \(\frac{5}{4x^{3}y^{4}}\)
- \(\frac{5x^{-3}}{4y^{4}}\)
- \(\frac{4y^{4}}{5x^{3}}\)
Expert · Level 6
View options
- (8)
- (9)
- (10)
- (12)
Expert · Level 6
View options
- \(5b^{2}\)
- \(5b^{-2}\)
- \(15b^{2}\)
- \(3b^{2}\)
Expert · Level 6
View options
- (250\sqrt{2})
- (125\sqrt{2})
- (100\sqrt{2})
- (50\sqrt{2})
Expert · Level 6
View options
- \(\frac{4721}{1600}\)
- \(\frac{89}{40}\)
- \(\frac{1600}{4721}\)
- \(\frac{39}{20}\)
Expert · Level 6
View options
- 2
- \\(\\frac{5}{3}\\)
- \\(\\frac{7}{3}\\)
- 3
Expert · Level 6
View options
- (13)
- (14)
- (15)
- (16)
Expert · Level 6
View options
- (1)
- (2)
- (\sqrt{m+n})
- (m+n)
Expert · Level 6
View options
- (1)
- (2)
- (3)
- (4)
If \(x=\sqrt{11}-\sqrt{6}\), what is the value of \(x^2+2\sqrt{66}\)?
Correct answer: C
Using the identity \((a-b)^2=a^2+b^2-2ab\), we get \(x^2=(\sqrt{11}-\sqrt{6})^2=11+6-2\sqrt{66}=17-2\sqrt{66}\). Therefore, \(x^2+2\sqrt{66}=17\). Option 5 results from incorrectly subtracting 6 from 11 while squaring. Exam tip: for \((a-b)^2\), the middle term is always \(-2ab\).
What is the simplified form of \(\left(\frac{9r^{-4}s^{3}}{81r^{2}s^{-5}}\right)^{-1}\)?
Correct answer: A
Inside, \(\frac{9r^{-4}s^{3}}{81r^{2}s^{-5}}=\frac{1}{9}r^{-6}s^{8}\). Raising to (-1) gives \(9r^{6}s^{-8}\).
If (16^{x}=1024) and (32^{y}=1024), what is the value of (x+y)?
Correct answer: A
Since (1024=2^{10}), (16^{x}=2^{4x}) gives (x=\frac{5}{2}), and (32^{y}=2^{5y}) gives (y=2). Hence the sum is (\frac{9}{2}).
If \(a>0\) and \(p,q\) are real numbers, which of the following laws of exponents is always correct?
Correct answer: A
When powers with the same positive base are multiplied, their exponents are added; hence \(a^p\cdot a^q=a^{p+q}\). In division, exponents are subtracted, not divided: \(a^p/a^q=a^{p-q}\). Exam tip: multiply → add exponents; divide → subtract.
What is the simplified form of (\frac{24^{3}}{2^{6}\cdot3^{2}})?
Correct answer: B
Since (24^{3}=(2^{3}\cdot3)^{3}=2^{9}\cdot3^{3}), division leaves (2^{3}\cdot3=24), so the correct value is not among the options.
If (s=4+\sqrt{17}), what is the value of (s^{2}-\frac{1}{s^{2}})?
Correct answer: A
Here (\frac{1}{s}=\sqrt{17}-4), so (s-\frac{1}{s}=8) and (s+\frac{1}{s}=2\sqrt{17}). Thus (s^{2}-\frac{1}{s^{2}}=16\sqrt{17}).
What is the simplified form of \(\left(\frac{125x^{-9}}{64y^{12}}\right)^{-\frac{1}{3}}\)?
Correct answer: A
We get \(\left(\frac{125x^{-9}}{64y^{12}}\right)^{\frac{1}{3}}=\frac{5x^{-3}}{4y^{4}}\). The power \(-\frac{1}{3}\) gives the reciprocal \(\frac{4x^{3}y^{4}}{5}\).
If \(x^{2}-\frac{1}{x^{2}}=60\) and \(x-\frac{1}{x}=6\), what is the value of \(x+\frac{1}{x}\)?
Correct answer: C
We use \(x^{2}-\frac{1}{x^{2}}=\left(x-\frac{1}{x}\right)\left(x+\frac{1}{x}\right)\). Thus \(60=6\left(x+\frac{1}{x}\right)\), so the value is (10).
What is the simplified form of the following expression, where \(b\ne0\)? \(\frac{6b^{-3}+9b^{-3}}{3b^{-5}}\)
Correct answer: A
Combining like terms gives \(6b^{-3}+9b^{-3}=15b^{-3}\). Therefore, \(\frac{15b^{-3}}{3b^{-5}}=5b^{-3-(-5)}=5b^2\), so option A is correct. Option B results from subtracting the exponents in the wrong order. Exam tip: when dividing powers with the same non-zero base, subtract the denominator exponent from the numerator exponent: \(b^m/b^n=b^{m-n}\).
If (\sqrt{x}=5\sqrt{2}), what is the value of (x^{\frac{3}{2}})?
Correct answer: A
From (\sqrt{x}=5\sqrt{2}), (x=50), and (x^{\frac{3}{2}}=x\sqrt{x}=50\cdot5\sqrt{2}=250\sqrt{2}). In exams, write (x^{\frac{3}{2}}) as (x\sqrt{x}).
What is the value of \(\left(\frac{5}{8}\right)^{-2}+\left(\frac{8}{5}\right)^{-2}\)?
Correct answer: A
Here \(\left(\frac{5}{8}\right)^{-2}=\frac{64}{25}\) and \(\left(\frac{8}{5}\right)^{-2}=\frac{25}{64}\). The sum is \(\frac{4096+625}{1600}=\frac{4721}{1600}\).
If \\( (7^x)^2\cdot 7^{x-1}=16807 \\), what is the value of \\(x\\)?
Correct answer: A
Using the laws of exponents, \\( (7^x)^2=7^{2x}\\), so the left-hand side becomes \\(7^{2x}\cdot7^{x-1}=7^{3x-1}\\). Also, \\(16807=7^5\\). Therefore, \\(3x-1=5\\), giving \\(3x=6\\) and hence \\(x=2\\). Exam tip: when multiplying powers with the same base, add their exponents.
What is the value of (\frac{\sqrt{363}-2\sqrt{147}+3\sqrt{75}}{\sqrt{3}})?
Correct answer: C
Here (\sqrt{363}=11\sqrt{3}), (2\sqrt{147}=14\sqrt{3}), and (3\sqrt{75}=15\sqrt{3}). The numerator is (12\sqrt{3}), so the value should be (12).
If (\frac{1}{\sqrt{m}+\sqrt{n}}=\sqrt{m}-\sqrt{n}) and (m>n>0), what is the value of (m-n)?
Correct answer: A
Multiplying both sides by (\sqrt{m}+\sqrt{n}) gives (1=m-n). In exams, apply the conjugate product directly.
If \(\left(3x^{-2}y^{3}\right)^{2}\cdot\left(9x^{4}y^{-1}\right)^{-1}\) is written as \(cx^{r}y^{s}\), what is the value of (c+r+s)?
Correct answer: B
The expression is \(9x^{-4}y^{6}\cdot\frac{1}{9}x^{-4}y=x^{-8}y^{7}\). Thus \(c=1\), \(r=-8\), \(s=7\), and \(c+r+s=0\).
QUIZ COMPLETE