Simplify the following indices: a) 18a2b/6ab b) y5xy2 c) 6⁻³ d) a2b⁻² / a3b⁻²

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ScanSolve AI Tutor · Accepted Answer

a) Simplify 18a³b² ÷ 6ab (Assuming the '=' sign in the question is a typo and should be '÷' for simplification.) Step 1: Rewrite the expression as a fraction. (18a³b²)/(6ab) Step 2: Divide the numerical coefficients. 18/6 = 3; Step 3: Apply the quotient rule for exponents (x^m ÷ xⁿ = x^(m−n)) for variables a and b. For a: a³⁻¹ = a²; For b: b²⁻¹ = b¹ = b Step 4: Combine the simplified parts. 3a²b The simplified expression is 3a²b. b) Simplify y³ × y² Step 1: Apply the product rule for exponents (x^m × xⁿ = x^(m+n)). y³⁺² Step 2: Add the exponents. y⁵ The simplified expression is y⁵. c) Simplify 6⁻³ Step 1: Apply the negative exponent rule (x⁻ⁿ = (1)/(xⁿ)). (1)/(6³) Step 2: Calculate the value of 6³. 6³ = 6 × 6 × 6 = 216 Step 3: Substitute the value back into the expression. 1/216 The simplified expression is 1/216. d) Simplify (a²b⁻²)/(a³b⁻²) Step 1: Apply the quotient rule for exponents for variable a. a²⁻³ = a⁻¹ Step 2: Apply the quotient rule for exponents for variable b. b⁻²⁻⁽⁻²⁾ = b⁻²⁺² = b⁰; Step 3: Apply the zero exponent rule (x⁰ = 1). b⁰ = 1 Step 4: Combine the simplified parts. a⁻¹ × 1 = a⁻¹ Step 5: Apply the negative exponent rule to express a⁻¹ with a positive exponent. 1/a The simplified expression is 1/a. e) Simplify 18a³b² ÷ 3ab Step 1: Rewrite the expression as a fraction. (18a³b²)/(3ab) Step 2: Divide the numerical coefficients. 18/3 = 6; Step 3: Apply the quotient rule for exponents (x^m ÷ xⁿ = x^(m−n)) for variables a and b. For a: a³⁻¹ = a²; For b: b²⁻¹ = b¹ = b Step 4: Combine the simplified parts. 6a²b The simplified expression is 6a²b.

Simplify the following indices: a) 18a2b/6ab b) y5xy2 c) 6⁻³ d) a2b⁻² / a3b⁻²

Answers

Step-by-step solution

Part a· 4 steps

Rewrite the expression as a fraction.

Divide the numerical coefficients.

Apply the quotient rule for exponents ( $x^m \div x^n = x^{m-n}$ ) for variables $a$ and $b$ .

Combine the simplified parts.

Part b· 2 steps
$\mathbf{y^5}$

Apply the product rule for exponents ( $x^m \times x^n = x^{m+n}$ ).

Add the exponents.

Part c· 3 steps
$\mathbf{\frac{1}{216}}$

Apply the negative exponent rule ( $x^{-n} = \frac{1}{x^n}$ ).

Calculate the value of $6^3$ .

Substitute the value back into the expression.

Part d· 5 steps
$\mathbf{\frac{1}{a}}$

Apply the quotient rule for exponents for variable $a$ .

Apply the quotient rule for exponents for variable $b$ .

Apply the zero exponent rule ( $x^0 = 1$ ).

Combine the simplified parts.

Apply the negative exponent rule to express $a^{-1}$ with a positive exponent.

Part e· 4 steps
$\mathbf{6a^2b}$

Rewrite the expression as a fraction.

Divide the numerical coefficients.

Apply the quotient rule for exponents ( $x^m \div x^n = x^{m-n}$ ) for variables $a$ and $b$ .

Combine the simplified parts.

Try one yourself

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More Mathematics Questions

Simplify the following indices: a) 18a2b/6ab b) y5xy2 c) 6⁻³ d) a2b⁻² / a3b⁻²

Answers

Step-by-step solution

Part a· 4 steps

Rewrite the expression as a fraction.

Divide the numerical coefficients.

Apply the quotient rule for exponents (xm÷xn=xm−nx^m \div x^n = x^{m-n}xm÷xn=xm−n) for variables aaa and bbb.

Combine the simplified parts.

Part b· 2 stepsy5\mathbf{y^5}y5

Apply the product rule for exponents (xm×xn=xm+nx^m \times x^n = x^{m+n}xm×xn=xm+n).

Add the exponents.

Part c· 3 steps1216\mathbf{\frac{1}{216}}2161​

Apply the negative exponent rule (x−n=1xnx^{-n} = \frac{1}{x^n}x−n=xn1​).

Calculate the value of 636^363.

Substitute the value back into the expression.

Part d· 5 steps1a\mathbf{\frac{1}{a}}a1​

Apply the quotient rule for exponents for variable aaa.

Apply the quotient rule for exponents for variable bbb.

Apply the zero exponent rule (x0=1x^0 = 1x0=1).

Combine the simplified parts.

Apply the negative exponent rule to express a−1a^{-1}a−1 with a positive exponent.

Part e· 4 steps6a2b\mathbf{6a^2b}6a2b

Rewrite the expression as a fraction.

Divide the numerical coefficients.

Apply the quotient rule for exponents (xm÷xn=xm−nx^m \div x^n = x^{m-n}xm÷xn=xm−n) for variables aaa and bbb.

Combine the simplified parts.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Apply the quotient rule for exponents ( $x^m \div x^n = x^{m-n}$ ) for variables $a$ and $b$ .

Part b· 2 steps
$\mathbf{y^5}$

Apply the product rule for exponents ( $x^m \times x^n = x^{m+n}$ ).

Part c· 3 steps
$\mathbf{\frac{1}{216}}$

Apply the negative exponent rule ( $x^{-n} = \frac{1}{x^n}$ ).

Calculate the value of $6^3$ .

Part d· 5 steps
$\mathbf{\frac{1}{a}}$

Apply the quotient rule for exponents for variable $a$ .

Apply the quotient rule for exponents for variable $b$ .

Apply the zero exponent rule ( $x^0 = 1$ ).

Apply the negative exponent rule to express $a^{-1}$ with a positive exponent.

Part e· 4 steps
$\mathbf{6a^2b}$

Apply the quotient rule for exponents ( $x^m \div x^n = x^{m-n}$ ) for variables $a$ and $b$ .