Match each expression (1–4) with the identically equal expression (A–E).
А
\(4x^2+4xy+y^2\) Б
\(4x^2-4xy+y^2\) В
\(4x^2-4xy-y^2\) Г
\(4x^2-3xy-y^2\) Д
\(4x^2-y^2\)
Solution
1) \((2x-y)^2=4x^2-4xy+y^2\) — B. 2) \((2x-y)(2x+y)=4x^2-y^2\) — E. 3) \((2x+y)^2=4x^2+4xy+y^2\) — A. 4) \((x-y)(4x+y)=4x^2-3xy-y^2\) — D. Matching: 1–B, 2–E, 3–A, 4–D.
Shapes
Grid
Shapes can be selected, moved, and resized.
2
Task
Numeric answer
Find the value of \(|y-2x|\) if \(4x^2-4xy+y^2=\dfrac94\).
Solution
\(4x^2-4xy+y^2=(2x-y)^2=\dfrac94\). Hence \(|2x-y|=\dfrac32=1.5\). Since \(|2x-y|=|y-2x|\), the answer is \(1.5\).
Shapes
Grid
Shapes can be selected, moved, and resized.
3
Task
One correct answer
If \(a<-2\), then \(1-|a+2|=\)
Solution
Since \(a<-2\), we have \(a+2<0\), so \(|a+2|=-(a+2)=-a-2\). Thus \(1-|a+2|=1-(-a-2)=a+3\). Answer: D.
Shapes
Grid
Shapes can be selected, moved, and resized.
4
Task
One correct answer
Write the expression \((3y-2)(3y+2)\) as a polynomial.
Solution
Using the difference of squares: \((3y-2)(3y+2)=(3y)^2-2^2=9y^2-4\). Answer: C.
Shapes
Grid
Shapes can be selected, moved, and resized.
5
Task
One correct answer
Find the value of \((5^{-5})^{-4}:(5^{-7})^{-3}\).
Solution
\((5^{-5})^{-4}=5^{20}\) and \((5^{-7})^{-3}=5^{21}\). Therefore \(5^{20}:5^{21}=5^{-1}=\dfrac15=0.2\). Answer: C.
Shapes
Grid
Shapes can be selected, moved, and resized.
6
Task
One correct answer
Simplify \(a-|a|\) if \(a<0\).
Solution
If \(a<0\), then \(|a|=-a\). Hence \(a-|a|=a-(-a)=2a\). Answer: A.
Shapes
Grid
Shapes can be selected, moved, and resized.
7
Task
Matching
For \(a>0\), match each expression (1–4) with the identically equal expression (A–E).
А
\(32a^{11}\) Б
\(2a^{\frac56}\) В
\(2a^{\frac65}\) Г
\(2a^{-1}\) Д
\(32a^{30}\)
Solution
1) \(\dfrac{2a^5}{a^6}=2a^{-1}\) — D. 2) \((2a)^5\cdot a^6=2^5a^{11}=32a^{11}\) — A. 3) \((2a^6)^5=2^5a^{30}=32a^{30}\) — E. 4) \(\sqrt[6]{64a^5}=64^{1/6}a^{5/6}=2a^{5/6}\) — B. Matching: 1–D, 2–A, 3–E, 4–B.
Shapes
Grid
Shapes can be selected, moved, and resized.
8
Task
One correct answer
Simplify \((a^6)^4:a^2\), where \(a\ne0\).
Solution
\((a^6)^4:a^2=a^{24}:a^2=a^{22}\). Answer: E.
Shapes
Grid
Shapes can be selected, moved, and resized.
9
Task
Numeric answer
Simplify \(2(a^2-5ab+4b^2)-3(2a^2-2ab+3b^2)\) and evaluate it for \(a=1.1\), \(b=0.8\).
Solution
\(2(a^2-5ab+4b^2)-3(2a^2-2ab+3b^2)=-4a^2-4ab-b^2=-(2a+b)^2\). For \(a=1.1\), \(b=0.8\): \(-(2\cdot1.1+0.8)^2=-3^2=-9\). Answer: \(-9\).
Shapes
Grid
Shapes can be selected, moved, and resized.
10
Task
Matching
Match the given expressions (1–4) with the expressions identically equal to them (A–E).
А
\(4a^2-b^2\) Б
\(4b^2-2ab+a^2\) В
\(2a^2+3ab-2b^2\) Г
\(4a^2+4ab+b^2\) Д
\(4b^2-4ab+a^2\)
Solution
1) \((2a+b)^2=4a^2+4ab+b^2\) — D. 2) \((2a-b)(b+2a)=(2a-b)(2a+b)=4a^2-b^2\) — A. 3) \((a-2b)^2=a^2-4ab+4b^2\) — E. 4) \((a+2b)(2a-b)=2a^2+3ab-2b^2\) — C. Matching: 1–D, 2–A, 3–E, 4–C.
Shapes
Grid
Shapes can be selected, moved, and resized.
11
Task
One correct answer
Write \((x+3)(2x-1)\) as a polynomial.
Solution
Expand: \((x+3)(2x-1)=2x^2-x+6x-3=2x^2+5x-3\). Answer: A.
Shapes
Grid
Shapes can be selected, moved, and resized.
12
Task
One correct answer
Factor the expression \(25x^2-1\).
Solution
Use the difference of squares: \(25x^2-1=(5x)^2-1^2=(5x-1)(5x+1)\). Answer: C.
Shapes
Grid
Shapes can be selected, moved, and resized.
13
Task
One correct answer
If \(2^a=\dfrac15\), then \(2^{6-a}=\)
Solution
\(2^{6-a}=\dfrac{2^6}{2^a}=\dfrac{64}{1/5}=320\). Answer: E.
Shapes
Grid
Shapes can be selected, moved, and resized.
14
Task
One correct answer
Simplify \(0.4x^2\cdot5x^3\).
Solution
\(0.4\cdot5=2\) and \(x^2\cdot x^3=x^5\). Thus \(0.4x^2\cdot5x^3=2x^5\). Answer: C.
Shapes
Grid
Shapes can be selected, moved, and resized.
15
Task
One correct answer
Factor \((a-1)^2-(b-1)^2\).
Solution
Using the difference of squares: \((a-1)^2-(b-1)^2=(a-b)(a+b-2)\). Answer: D.
Shapes
Grid
Shapes can be selected, moved, and resized.
16
Task
One correct answer
Choose the expression identically equal to \(x^2+4\).
Solution
For E: \((x-2)^2+4x=x^2-4x+4+4x=x^2+4\). Answer: E.
Shapes
Grid
Shapes can be selected, moved, and resized.
17
Task
Matching
For \(a=15\), match each expression (1–4) with the correct statement about its value (A–E).
А
is less than 20 Б
is a prime number В
is even Г
is divisible by 3 Д
is divisible by 5
Solution
For \(a=15\): 1) \(\dfrac73\cdot15=35\), divisible by 5 — E. 2) \(2\cdot15-1=29\), prime — B. 3) \(15^2+12\cdot15+36=441\), divisible by 3 — D. 4) \(15^2+13^2=394\), even — C. Matching: 1–E, 2–B, 3–D, 4–C.
Shapes
Grid
Shapes can be selected, moved, and resized.
18
Task
One correct answer
Simplify \(\dfrac{2a+2}{2}\).
Solution
\(\dfrac{2a+2}{2}=\dfrac{2(a+1)}2=a+1\). Answer: C.
Shapes
Grid
Shapes can be selected, moved, and resized.
19
Task
One correct answer
Simplify \(a(a+2b)-(a+b)^2\).
Solution
Expand: \(a(a+2b)-(a+b)^2=a^2+2ab-(a^2+2ab+b^2)=-b^2\). Answer: C.
Shapes
Grid
Shapes can be selected, moved, and resized.
20
Task
One correct answer
If \(x^2-y^2=7\) and \(3x+3y=63\), then \(x-y=\)
Solution
From \(3x+3y=63\), \(x+y=21\). Since \(x^2-y^2=(x-y)(x+y)=7\), we get \((x-y)\cdot21=7\), so \(x-y=\dfrac13\). Answer: E.
Shapes
Grid
Shapes can be selected, moved, and resized.
21
Task
One correct answer
If \(2^x=3\), then \(4^{x+1}=\)
Solution
\(4^{x+1}=4\cdot4^x=4\cdot(2^x)^2=4\cdot3^2=36\). Answer: D.
Shapes
Grid
Shapes can be selected, moved, and resized.
22
Task
One correct answer
Simplify \(0.8b^9:(8b^3)\), where \(b\ne0\).
Solution
\(0.8b^9:(8b^3)=\dfrac{0.8}{8}b^{9-3}=0.1b^6\). Answer: A.
Shapes
Grid
Shapes can be selected, moved, and resized.
23
Task
One correct answer
Simplify \(2a-(3b-2a)\).
Solution
Expand the brackets: \(2a-(3b-2a)=2a-3b+2a=4a-3b\). Answer: B.
Shapes
Grid
Shapes can be selected, moved, and resized.
24
Task
Matching
A point with coordinate \(a\) is marked on the \(x\)-axis as shown.
Match each expression (1–3) with the point on the \(x\)-axis (A–E) whose coordinate equals the value of the expression.
А
\(M\) Б
\(L\) В
\(P\) Г
\(K\) Д
\(N\)
Solution
From the scale, \(a\approx0.75\). Then \(-2a\approx-1.5\) — point \(K\) (D); \(3^a\approx3^{0.75}\approx2.28\) — point \(P\) (C); \(|a-1|\approx0.25\) — point \(M\) (A). Matching: 1–D, 2–C, 3–A.
Shapes
Grid
Shapes can be selected, moved, and resized.
25
Task
One correct answer
Evaluate \(\left(\dfrac13\right)^{-2}\).
Solution
Using the negative-exponent rule: \(\left(\dfrac13\right)^{-2}=3^2=9\). Answer: E.
Shapes
Grid
Shapes can be selected, moved, and resized.
26
Task
One correct answer
Factor \((x+y)^2-9x^2\).
Solution
Use the difference of squares: \((x+y)^2-(3x)^2=(x+y-3x)(x+y+3x)=(-2x+y)(4x+y)\). Answer: C.
Shapes
Grid
Shapes can be selected, moved, and resized.
27
Task
One correct answer
Simplify \(x+2(x-2)\).
Solution
\(x+2(x-2)=x+2x-4=3x-4\). Answer: A.
Shapes
Grid
Shapes can be selected, moved, and resized.
28
Task
One correct answer
Choose the expression identically equal to \((2x-3)^2+12x\).
Solution
\((2x-3)^2+12x=4x^2-12x+9+12x=4x^2+9\). Answer: B.
Shapes
Grid
Shapes can be selected, moved, and resized.
29
Task
One correct answer
Simplify \((a-4)^2-a^2\).
Solution
\((a-4)^2-a^2=a^2-8a+16-a^2=-8a+16\). Answer: A.
Shapes
Grid
Shapes can be selected, moved, and resized.
30
Task
One correct answer
Evaluate \(\dfrac{15^3}{3^2}\).
Solution
\(\dfrac{15^3}{3^2}=\dfrac{(5\cdot3)^3}{3^2}=5^3\cdot3^{3-2}=125\cdot3=375\). Answer: D.
Shapes
Grid
Shapes can be selected, moved, and resized.
31
Task
One correct answer
Simplify \(\dfrac{3m-2n}{8}-\dfrac{3m}{8}\).
Solution
\(\dfrac{3m-2n}{8}-\dfrac{3m}{8}=\dfrac{-2n}{8}=-\dfrac n4\). Answer: A.
Shapes
Grid
Shapes can be selected, moved, and resized.
32
Task
One correct answer
Simplify \(2(x+5y)-(4y-7x)\).
Solution
\(2(x+5y)-(4y-7x)=2x+10y-4y+7x=9x+6y\). Answer: D.
Shapes
Grid
Shapes can be selected, moved, and resized.
33
Task
One correct answer
Simplify \(3(1-x)(1+x)\).
Solution
Using the difference of squares, \((1-x)(1+x)=1-x^2\), so \(3(1-x)(1+x)=3-3x^2\). Answer: A.
Shapes
Grid
Shapes can be selected, moved, and resized.
34
Task
Matching
Match each expression (1–3) with its value (A–E), given \(a=\dfrac{25}{4}\).
А
\(2\dfrac12\) Б
\(\dfrac4{25}\) В
\(3\dfrac12\) Г
\(4\dfrac16\) Д
\(-3\dfrac12\)
Solution
1) \(\dfrac{2a}{3}=\dfrac{25}{6}=4\dfrac16\) — D. 2) \(\dfrac1a=\dfrac4{25}\) — B. 3) \(|9-2a|=|9-25/2|=7/2=3\dfrac12\) — C. Matching: 1–D, 2–B, 3–C.
Shapes
Grid
Shapes can be selected, moved, and resized.
35
Task
One correct answer
Simplify \((-2a^{-3}b)^{-2}\).
Solution
\((-2a^{-3}b)^{-2}=(-2)^{-2}(a^{-3})^{-2}b^{-2}=\dfrac{a^6}{4b^2}\). Answer: C.
