Solve the equation rac92x−33x+3 = 1. 9. Given that 3 imes 91+x = 27^-x, find x. 10. Simplify rac135n imes…

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Solve the equation rac92x−33x+3 = 1. 9. Given that 3 imes 91+x = 27^-x, find x. 10. Simplify rac135n imes 9ⁿ−1 imes 27ⁿ+1.

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9. Given that 3 × 9^(1+x) = 27^(−x), find x. Step 1: Express all terms with the same base, which is 3. 3¹ × (3²)^(1+x) = (3³)^(−x); Step 2: Apply the exponent rule (a^m)ⁿ = a^(mn). 3¹ × 3^(2(1+x)) = 3^(3(−x)); 3¹ × 3^(2+2x) = 3^(−3x); Step 3: Apply the exponent rule a^m × aⁿ = a^(m+n) on the left side. 3^(1 + (2+2x)) = 3^(−3x); 3^(3+2x) = 3^(−3x) Step 4: Equate the exponents since the bases are the same. 3+2x = −3x Step 5: Solve for x. 3 = −3x − 2x; 3 = −5x; x = −3/5 The value of x is −3/5. 10. Simplify (1)/(3⁵ⁿ) × 9ⁿ⁻¹ × 27ⁿ⁺¹. Step 1: Express all terms with the same base, which is 3. (1)/(3⁵ⁿ) × (3²)ⁿ⁻¹ × (3³)ⁿ⁺¹ Step 2: Apply the exponent rules (1)/(a^m) = a^(−m) and (a^m)ⁿ = a^(mn). 3⁻⁵ⁿ × 3²⁽ⁿ⁻¹⁾ × 3³⁽ⁿ⁺¹⁾ 3⁻⁵ⁿ × 3²ⁿ⁻² × 3³ⁿ⁺³ Step 3: Apply the exponent rule a^m × aⁿ × a^p = a^(m+n+p). 3⁻⁵ⁿ⁺⁽²ⁿ⁻²⁾⁺⁽³ⁿ⁺³⁾ 3⁻⁵ⁿ⁺²ⁿ⁻²⁺³ⁿ⁺³ Step 4: Combine the exponents. 3⁽⁻⁵ⁿ⁺²ⁿ⁺³ⁿ⁾⁺⁽⁻²⁺³⁾ 3⁰ⁿ⁺¹ 3¹ The simplified expression is 3. 11. Solve the equation (9^(2x−3))/(3^(x+3)) = 1. Step 1: Express all terms with the same base, which is 3. ((3²)^(2x−3))/(3^(x+3)) = 3⁰; Step 2: Apply the exponent rule (a^m)ⁿ = a^(mn). (3^(2(2x−3)))/(3^(x+3)) = 3⁰; (3^(4x−6))/(3^(x+3)) = 3⁰; Step 3: Apply the exponent rule (a^m)/(aⁿ) = a^(m−n). 3^((4x−6) − (x+3)) = 3⁰; 3^(4x−6−x−3) = 3⁰; 3^(3x−9) = 3⁰ Step 4: Equate the exponents since the bases are the same. 3x−9 = 0 Step 5: Solve for x. 3x = 9; x = 9/3; x = 3 The value of x is 3. 12. (a) Solve (1)/(81^((x−2))) = 27^((1−x)). Step 1: Express all terms with the same base, which is 3. (1)/((3⁴)^(x−2)) = (3³)^(1−x); Step 2: Apply the exponent rules (1)/(a^m) = a^(−m) and (a^m)ⁿ = a^(mn). (3⁴)^(−(x−2)) = 3^(3(1−x)); 3^(−4(x−2)) = 3^(3−3x); 3^(−4x+8) = 3^(3−3x) Step 3: Equate the exponents since the bases are the same. −4x+8 = 3−3x Step 4: Solve for x. 8−3 = −3x+4x; 5 = x The value of x is 5. 13. Evaluate 2 ÷ (64/125)^(−2/3). Step 1: Simplify the term with the negative exponent using (a/b)^(−m) = (b/a)^m. 2 ÷ (125/64)^(2/3) Step 2: Apply the fractional exponent. Recall that a^(m/n) = (√[n]a)^m. 2 ÷ (√[3]125/64)² Step 3: Calculate the cube roots. Note that 125 = 5³ and 64 = 4³. 2 ÷ (√[3]125√[3]64)² 2 ÷ (5/4)² Step 4: Square the fraction. 2 ÷ (5²)/(4²) 2 ÷ 25/16 Step 5: Perform the division by multiplying by the reciprocal. 2 × 16/25 32/25 The evaluated expression is 32/25. 14. (a) Evaluate and express your answer in standard form; (4.56 × 3.6)/(0.12). Step 1: Multiply the numbers in the numerator. 4.56 × 3.6 = 16.416 Step 2: Divide the result by the denominator. 16.416/0.12 = 136.8 Step 3: Express the answer in standard form (scientific notation). A number in standard form is written as a × 10ⁿ, where 1 ≤ |a| < 10. 136.8 = 1.368 × 10² The evaluated expression in standard form is 1.368 × 10². 14. (b) Without using mathematical tables or calculator, evaluate (73.8)² − (26.2)². Step 1: Use the difference of squares formula: a² − b² = (a−b)(a+b). Here, a = 73.8 and b = 26.2. Step 2: Substitute the values into the formula. (73.8 − 26.2)(73.8 + 26.2) Step 3: Perform the subtraction and addition. 73.8 − 26.2 = 47.6; 73.8 + 26.2 = 100.0 Step 4: Multiply the results. 47.6 × 100 = 4760 The evaluated expression is 4760. 15. (a) Simplify (0.016 × 0.084)/(0.48). Leaving your answer in standard form. Step 1: Convert the decimals to fractions or powers of 10 to simplify calculations. 0.016 = 16/1000 = 16 × 10⁻³; 0.084 = 84/1000 = 84 × 10⁻³; 0.48 = 48/100 = 48 × 10⁻² Step 2: Substitute these into the expression. ((16 × 10⁻³) × (84 × 10⁻³))/(48 × 10⁻²) Step 3: Multiply the numbers in the numerator and combine powers of 10. ((16 × 84) × (10⁻³ × 10⁻³))/(48 × 10⁻²) (1344 × 10⁻⁶)/(48 × 10⁻²) Step 4: Divide the numerical parts and the powers of 10. 1344/48 × (10⁻⁶)/(10⁻²) 28 × 10⁻⁶⁻⁽⁻²⁾ 28 × 10⁻⁶⁺² 28 × 10⁻⁴ Step 5: Express the answer in standard form (a × 10ⁿ where $1 ≤ |

Solve the equation rac92x−33x+3 = 1. 9. Given that 3 imes 91+x = 27^-x, find x. 10. Simplify rac135n imes 9ⁿ-1 imes 27ⁿ+1.

Step-by-step solution

Express all terms with the same base, which is 3.

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Apply the exponent rule $a^m \times a^n = a^{m+n}$ on the left side.

Equate the exponents since the bases are the same.

Solve for $x$ .

Express all terms with the same base, which is 3.

Apply the exponent rules $\frac{1}{a^m} = a^{-m}$ and $(a^m)^n = a^{mn}$ .

Apply the exponent rule $a^m \times a^n \times a^p = a^{m+n+p}$ .

Combine the exponents.

Express all terms with the same base, which is 3.

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .

Equate the exponents since the bases are the same.

Solve for $x$ .

Express all terms with the same base, which is 3.

Apply the exponent rules $\frac{1}{a^m} = a^{-m}$ and $(a^m)^n = a^{mn}$ .

Equate the exponents since the bases are the same.

Solve for $x$ .

Simplify the term with the negative exponent using $(a/b)^{-m} = (b/a)^m$ .

Apply the fractional exponent.

Calculate the cube roots.

Square the fraction.

Perform the division by multiplying by the reciprocal.

Multiply the numbers in the numerator.

Divide the result by the denominator.

Express the answer in standard form (scientific notation).

Use the difference of squares formula

Substitute the values into the formula.

Perform the subtraction and addition.

Multiply the results.

Convert the decimals to fractions or powers of 10 to simplify calculations.

Substitute these into the expression.

Multiply the numbers in the numerator and combine powers of 10.

Divide the numerical parts and the powers of 10.

Express the answer in standard form ( $a \times 10^n$ where $1 \le |

Still stuck on this one?

More Mathematics Questions

Solve the equation rac92x−33x+3 = 1. 9. Given that 3 imes 91+x = 27^-x, find x. 10. Simplify rac135n imes 9ⁿ-1 imes 27ⁿ+1.

Step-by-step solution

Express all terms with the same base, which is 3.

Apply the exponent rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Apply the exponent rule am×an=am+na^m \times a^n = a^{m+n}am×an=am+n on the left side.

Equate the exponents since the bases are the same.

Solve for xxx.

Express all terms with the same base, which is 3.

Apply the exponent rules 1am=a−m\frac{1}{a^m} = a^{-m}am1​=a−m and (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Apply the exponent rule am×an×ap=am+n+pa^m \times a^n \times a^p = a^{m+n+p}am×an×ap=am+n+p.

Combine the exponents.

Express all terms with the same base, which is 3.

Apply the exponent rule (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Apply the exponent rule aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n.

Equate the exponents since the bases are the same.

Solve for xxx.

Express all terms with the same base, which is 3.

Apply the exponent rules 1am=a−m\frac{1}{a^m} = a^{-m}am1​=a−m and (am)n=amn(a^m)^n = a^{mn}(am)n=amn.

Equate the exponents since the bases are the same.

Solve for xxx.

Simplify the term with the negative exponent using (a/b)−m=(b/a)m(a/b)^{-m} = (b/a)^m(a/b)−m=(b/a)m.

Apply the fractional exponent.

Calculate the cube roots.

Square the fraction.

Perform the division by multiplying by the reciprocal.

Multiply the numbers in the numerator.

Divide the result by the denominator.

Express the answer in standard form (scientific notation).

Use the difference of squares formula

Substitute the values into the formula.

Perform the subtraction and addition.

Multiply the results.

Convert the decimals to fractions or powers of 10 to simplify calculations.

Substitute these into the expression.

Multiply the numbers in the numerator and combine powers of 10.

Divide the numerical parts and the powers of 10.

Express the answer in standard form (a×10na \times 10^na×10n where $1 \le |

Still stuck on this one?

More Mathematics Questions

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Apply the exponent rule $a^m \times a^n = a^{m+n}$ on the left side.

Solve for $x$ .

Apply the exponent rules $\frac{1}{a^m} = a^{-m}$ and $(a^m)^n = a^{mn}$ .

Apply the exponent rule $a^m \times a^n \times a^p = a^{m+n+p}$ .

Apply the exponent rule $(a^m)^n = a^{mn}$ .

Apply the exponent rule $\frac{a^m}{a^n} = a^{m-n}$ .

Solve for $x$ .

Apply the exponent rules $\frac{1}{a^m} = a^{-m}$ and $(a^m)^n = a^{mn}$ .

Solve for $x$ .

Simplify the term with the negative exponent using $(a/b)^{-m} = (b/a)^m$ .

Express the answer in standard form ( $a \times 10^n$ where $1 \le |