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Mathematics Operations on real numbers and the laws of exponents MCQ Questions for Class 10

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Operations on real numbers and the laws of exponents Practice Questions

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Question 461/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

\(\frac{7^{4}\cdot49^{-1}}{343^{-2}}\) का सरल मान क्या है?

What is the simplified value of \(\frac{7^{4}\cdot49^{-1}}{343^{-2}}\)?

Explanation opens after your attempt
Correct Answer

A. \(7^{8}\)

Explanation

Simple Explanation

\(49^{-1}=7^{-2}\) और \(343^{-2}=7^{-6}\), इसलिए \(\frac{7^{4}\cdot7^{-2}}{7^{-6}}=7^{8}\)। परीक्षा में ऋणात्मक घात से भाग करने पर घात जुड़ती है। / Here \(49^{-1}=7^{-2}\) and \(343^{-2}=7^{-6}\), so \(\frac{7^{4}\cdot7^{-2}}{7^{-6}}=7^{8}\). In exams, dividing by a negative power adds the exponent.

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Question 462/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

\(\sqrt{98}-\sqrt{72}+\sqrt{32}-\sqrt{18}\) का सरल रूप क्या है?

What is the simplified form of \(\sqrt{98}-\sqrt{72}+\sqrt{32}-\sqrt{18}\)?

Explanation opens after your attempt
Correct Answer

A. \(2\sqrt{2}\)

Explanation

Simple Explanation

\(\sqrt{98}=7\sqrt{2}\), \(\sqrt{72}=6\sqrt{2}\), \(\sqrt{32}=4\sqrt{2}\), और \(\sqrt{18}=3\sqrt{2}\), इसलिए मान \(2\sqrt{2}\) है। परीक्षा में समान करणी पदों को ही जोड़ें। / We have \(\sqrt{98}=7\sqrt{2}\), \(\sqrt{72}=6\sqrt{2}\), \(\sqrt{32}=4\sqrt{2}\), and \(\sqrt{18}=3\sqrt{2}\), so the value is \(2\sqrt{2}\). In exams, combine only like radicals.

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Question 463/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

यदि \(3^{x}\cdot9^{x-1}=243\), तो (x) का मान क्या है?

If \(3^{x}\cdot9^{x-1}=243\), what is the value of (x)?

Explanation opens after your attempt
Correct Answer

B. \(\frac{7}{3}\)

Explanation

Simple Explanation

\(9^{x-1}=3^{2x-2}\), इसलिए कुल घात (x+2x-2=3x-2) है। \(243=3^{5}\) से (3x-2=5) और \(x=\frac{7}{3}\)। / Since \(9^{x-1}=3^{2x-2}\), the total exponent is (x+2x-2=3x-2). From \(243=3^{5}\), (3x-2=5), so \(x=\frac{7}{3}\).

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Question 464/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

\(\frac{x^{-2}-y^{-2}}{x^{-1}-y^{-1}}\) का सरल रूप क्या है, जहाँ \(x\neq0\), \(y\neq0\), और \(x\neq y\)?

What is the simplified form of \(\frac{x^{-2}-y^{-2}}{x^{-1}-y^{-1}}\), where \(x\neq0\), \(y\neq0\), and \(x\neq y\)?

Explanation opens after your attempt
Correct Answer

A. \(\frac{x+y}{xy}\)

Explanation

Simple Explanation

अंश \(\frac{y^{2}-x^{2}}{x^{2}y^{2}}\) और हर \(\frac{y-x}{xy}\) है, इसलिए भाग देने पर \(\frac{x+y}{xy}\) मिलता है। परीक्षा में ऋणात्मक घातों को भिन्न में बदलें। / The numerator is \(\frac{y^{2}-x^{2}}{x^{2}y^{2}}\) and the denominator is \(\frac{y-x}{xy}\), so division gives \(\frac{x+y}{xy}\). In exams, convert negative powers to fractions.

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Question 465/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

यदि \(A=9+4\sqrt{5}\), तो \(\sqrt{A}\) का सरल रूप क्या है?

If \(A=9+4\sqrt{5}\), what is the simplified form of \(\sqrt{A}\)?

Explanation opens after your attempt
Correct Answer

A. \(2+\sqrt{5}\)

Explanation

Simple Explanation

क्योंकि (\(2+\sqrt{5}\)^{2}=4+5+4\sqrt{5}=9+4\sqrt{5}), इसलिए \(\sqrt{A}=2+\sqrt{5}\)। परीक्षा में पूर्ण वर्ग करणी को पहचानें। / Because (\(2+\sqrt{5}\)^{2}=4+5+4\sqrt{5}=9+4\sqrt{5}), \(\sqrt{A}=2+\sqrt{5}\). In exams, recognize a perfect-square surd form.

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Question 466/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

(\left\(\frac{4x^{3}y^{-2}}{2x^{-1}y^{4}}\right\)^{2}\cdot\frac{y^{12}}{x^{4}}) का सरल रूप क्या है?

What is the simplified form of (\left\(\frac{4x^{3}y^{-2}}{2x^{-1}y^{4}}\right\)^{2}\cdot\frac{y^{12}}{x^{4}})?

Explanation opens after your attempt
Correct Answer

A. \(4x^{4}\)

Explanation

Simple Explanation

अंदर \(\frac{4x^{3}y^{-2}}{2x^{-1}y^{4}}=2x^{4}y^{-6}\), इसका वर्ग \(4x^{8}y^{-12}\) है। फिर \(\frac{y^{12}}{x^{4}}\) से गुणा करने पर \(4x^{4}\) मिलता है। / Inside, \(\frac{4x^{3}y^{-2}}{2x^{-1}y^{4}}=2x^{4}y^{-6}\), and its square is \(4x^{8}y^{-12}\). Multiplying by \(\frac{y^{12}}{x^{4}}\) gives \(4x^{4}\).

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Question 467/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

यदि \(2^{x}+2^{x+1}+2^{x+2}=112\), तो (x) का मान क्या है?

If \(2^{x}+2^{x+1}+2^{x+2}=112\), what is the value of (x)?

Explanation opens after your attempt
Correct Answer

B. (4)

Explanation

Simple Explanation

सामान्य पद \(2^{x}\) लेने पर (2^{x}(1+2+4)=112), इसलिए \(7\cdot2^{x}=112\)। इससे \(2^{x}=16\) और (x=4)। / Factoring \(2^{x}\), we get (2^{x}(1+2+4)=112), so \(7\cdot2^{x}=112\). Thus \(2^{x}=16\) and (x=4).

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Question 468/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

(\left\(27^{\frac{2}{3}}\right\)^{-1}\cdot\left\(81^{\frac{3}{4}}\right\)) का मान क्या है?

What is the value of (\left\(27^{\frac{2}{3}}\right\)^{-1}\cdot\left\(81^{\frac{3}{4}}\right\))?

Explanation opens after your attempt
Correct Answer

A. (3)

Explanation

Simple Explanation

\(27^{\frac{2}{3}}=9\), इसलिए पहला पद \(\frac{1}{9}\) है, और \(81^{\frac{3}{4}}=27\)। गुणनफल (3) है। / Here \(27^{\frac{2}{3}}=9\), so the first factor is \(\frac{1}{9}\), and \(81^{\frac{3}{4}}=27\). The product is (3).

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Question 469/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

यदि \(r=\sqrt{10}+\sqrt{2}\), तो \(r^{2}-4\sqrt{5}\) का मान क्या है?

If \(r=\sqrt{10}+\sqrt{2}\), what is the value of \(r^{2}-4\sqrt{5}\)?

Explanation opens after your attempt
Correct Answer

A. (12)

Explanation

Simple Explanation

\(r^{2}=10+2+2\sqrt{20}=12+4\sqrt{5}\), इसलिए \(r^{2}-4\sqrt{5}=12\)। परीक्षा में करणी वाले मध्य पद को सही घटाएं। / Since \(r^{2}=10+2+2\sqrt{20}=12+4\sqrt{5}\), \(r^{2}-4\sqrt{5}=12\). In exams, subtract the radical middle term correctly.

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Question 470/600 Expert Mathematics Polynomials Operations on real numbers and the laws of exponents Class 10 Level 43

(\frac{\(x^{4}-16\)}{\(x^{2}-4\)}) का सरल रूप क्या है, जहाँ \(x\neq2\) और \(x\neq-2\)?

What is the simplified form of (\frac{\(x^{4}-16\)}{\(x^{2}-4\)}), where \(x\neq2\) and \(x\neq-2\)?

Explanation opens after your attempt
Correct Answer

A. \(x^{2}+4\)

Explanation

Simple Explanation

(x^{4}-16=\(x^{2}-4\)\(x^{2}+4\)), इसलिए समान गुणनखंड कटने पर \(x^{2}+4\) बचता है। परीक्षा में वर्गों के अंतर को पहचानें। / Since (x^{4}-16=\(x^{2}-4\)\(x^{2}+4\)), cancelling the common factor leaves \(x^{2}+4\). In exams, recognize the difference of squares.

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