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Surds and Indices

23 questions

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Question 21Question →

What is the simplified value of 7+43−3\sqrt{7 + 4\sqrt{3}} - \sqrt{3}?

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Answer: 2

Answer

The simplified value is 2.
Expressing 7+437 + 4\sqrt{3} as (2+3)2(2 + \sqrt{3})^2 allows the square root to simplify directly to 2+32 + \sqrt{3}. Subtracting 3\sqrt{3} leaves the exact numerical answer 2.

Step-by-Step Solution

1
Rewrite the expression under the square root as a perfect square of a binomial.
7+43=22+(3)2+2(2)(3)=(2+3)27 + 4\sqrt{3} = 2^2 + (\sqrt{3})^2 + 2(2)(\sqrt{3}) = (2 + \sqrt{3})^2
Using the identity (a+b)2=a2+b2+2ab(a+b)^2 = a^2 + b^2 + 2ab, setting a=2a = 2 and b=3b = \sqrt{3} yields a2+b2=4+3=7a^2 + b^2 = 4 + 3 = 7 and 2ab=432ab = 4\sqrt{3}.
2
Evaluate the square root of the perfect square.
(2+3)2=2+3\sqrt{(2 + \sqrt{3})^2} = 2 + \sqrt{3}
The principal square root of a positive squared expression x2\sqrt{x^2} is xx.
3
Perform the subtraction indicated in the stem.
(2+3)−3=2(2 + \sqrt{3}) - \sqrt{3} = 2
The radical terms 3\sqrt{3} and −3-\sqrt{3} cancel out, leaving the integer 2.

Key Concept

Simplification of Nested Surds
Question 22Question →

If 2x+3⋅42x−18x+1=64\frac{2^{x+3} \cdot 4^{2x-1}}{8^{x+1}} = 64, what is the value of (x+2)2(x + 2)^2?

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Answer: 36

Answer

The value of (x+2)2(x + 2)^2 is 36.
Converting all terms to base 2 simplifies the equation to 22x−2=262^{2x-2} = 2^6. Equating the powers gives 2x−2=62x - 2 = 6, so x=4x = 4. Substituting x=4x = 4 into (x+2)2(x + 2)^2 yields (4+2)2=36(4 + 2)^2 = 36.

Step-by-Step Solution

1
Convert all exponential terms to base 2
Numerator term 42x−1=24x−24^{2x-1} = 2^{4x-2}, denominator term 8x+1=23x+38^{x+1} = 2^{3x+3}, and right-hand side 64=2664 = 2^6.
To combine powers using index laws, all expressions must share a common base.
2
Simplify the left-hand side expression using exponent laws
2x+3⋅24x−223x+3=25x+123x+3=2(5x+1)−(3x+3)=22x−2\frac{2^{x+3} \cdot 2^{4x-2}}{2^{3x+3}} = \frac{2^{5x+1}}{2^{3x+3}} = 2^{(5x+1)-(3x+3)} = 2^{2x-2}.
Apply product law am⋅an=am+na^m \cdot a^n = a^{m+n} and quotient law aman=am−n\frac{a^m}{a^n} = a^{m-n}.
3
Solve for the variable x
22x−2=26  ⟹  2x−2=6  ⟹  x=42^{2x-2} = 2^6 \implies 2x - 2 = 6 \implies x = 4.
When bases are equal, exponents must be equal.
4
Evaluate the requested target expression
(4+2)2=62=36(4 + 2)^2 = 6^2 = 36.
Substitute the calculated value of x=4x = 4 into (x+2)2(x + 2)^2.

Key Concept

Laws of Indices and Exponential Equations
Estimated Time:1m 30s
Question 23Question →

If the equation x+x−9=9\sqrt{x} + \sqrt{x - 9} = 9 holds true, what is the exact value of xx?

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Answer: 25

Answer

25
The exact value of xx is 25. By moving one radical to the right side and squaring both sides, we eliminate one square root. Simplifying and isolating the remaining square root allows us to square both sides a second time, revealing the final value. Alternatively, using the conjugate property of surds: multiplying both sides of the identity (x)2−(x−9)2=9(\sqrt{x})^2 - (\sqrt{x-9})^2 = 9 by their difference gives (x−x−9)(x+x−9)=9(\sqrt{x} - \sqrt{x-9})(\sqrt{x} + \sqrt{x-9}) = 9. Since the sum is 9, the difference must be 1 (i.e., x−x−9=1\sqrt{x} - \sqrt{x-9} = 1). Adding this back to the original equation yields 2x=102\sqrt{x} = 10, so x=5\sqrt{x} = 5 and x=25x = 25.

Step-by-Step Solution

1
Isolate one of the square root terms on one side of the equation.
x=9−x−9\sqrt{x} = 9 - \sqrt{x - 9}
Isolating a radical makes it easier to eliminate it by squaring both sides.
2
Square both sides of the equation and expand the right side.
x=81−18x−9+(x−9)x = 81 - 18\sqrt{x - 9} + (x - 9)
Squaring eliminates the isolated radical. The right side is expanded using the algebraic identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
3
Simplify the equation by canceling xx from both sides and combining constant terms.
x=x+72−18x−9⇒18x−9=72x = x + 72 - 18\sqrt{x - 9} \Rightarrow 18\sqrt{x - 9} = 72
Combining like terms simplifies the equation, leaving only a single radical expression.
4
Divide by 18 and square both sides one final time to solve for xx.
x−9=4⇒x−9=16⇒x=25\sqrt{x - 9} = 4 \Rightarrow x - 9 = 16 \Rightarrow x = 25
Isolating the final radical and squaring removes the remaining root, yielding a simple linear equation for xx.

Key Concept

Solving radical equations and applying algebraic identities with surds.
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