Respuesta :

Answer:

  • x=5  → [tex]\sqrt{(x-1)^3}=8[/tex]
  • x=19 → [tex]\sqrt[4]{(x-3)^5}=32[/tex]
  • x=29 → [tex]\sqrt{(x-4)^3}=125[/tex]
  • x=6 → [tex]\sqrt[3]{(x+2)^4}=16[/tex]

Step-by-step explanation:

First tile:

[tex]\sqrt{(x-1)^3}=8[/tex]

When we put x=5 we obtain:

[tex]\sqrt{(5-1)^3}=8\\\\\sqrt{4^3}=8\\\\\sqrt{64}=8\\\\\sqrt{8^2}=8\\\\8=8[/tex]

Hence, the first tile must be dragged to x=5

Second tile:

[tex]\sqrt[4]{(x-3)^5}=32\\\\(x-3)^5=(32)^4\\\\(x-3)^5=(2^5)^4[/tex]

Now when x=19

we have:

[tex](19-3)^5=(2^5)^4\\\\(16)^5=2^{20}\\\\(2^4)^5=2^{20}\\\\2^{20}=2^{20}[/tex]

( Since:

[tex](a^m)^n=a^{mn}[/tex] )

Third tile:

[tex]\sqrt{(x-4)^3}=125\\\\(x-4)^3=(125)^2\\\\Since\ on\ squaring\ both\ side\ of\ the\ equation\\\\(x-4)^3=(5^3)^2\\\\(x-4)^3=5^6[/tex]

when x=29 we have:

[tex](29-4)^3=5^6\\\\(25)^3=5^6\\\\(5^2)^3=5^6\\\\5^6=5^6[/tex]

Fourth tile:

[tex]\sqrt[3]{(x+2)^4}=16\\\\On\ cubing\ both\ side\ of\ the\ equation\ we\ get:\\\\(x+2)^4=(16)^3\\\\(x+2)^4=(2^4)^3\\\\(x+2)^4=2^{12}[/tex]

when x=6 we have:

[tex]8^4=2^{12}\\\\(2^3)^4=2^{12}\\\\2^{12}=2^{12}[/tex]

johanrusli

The matching of each radical eqaution with its solutions is as follows.

[tex]\large {\boxed {x = 19} }[/tex] → [tex]\sqrt[4]{(x-3)^5} = 32[/tex]

[tex]\large {\boxed {x = 6} }[/tex] → [tex]\sqrt[3]{(x+2)^4} = 16[/tex]

[tex]\large {\boxed {x = 29} }[/tex] → [tex]\sqrt{(x-4)^3} = 125[/tex]

[tex]\large {\boxed {x = 5} }[/tex] → [tex]\sqrt{(x-1)^3} = 8[/tex]

Further explanation

Let's recall following formula about Exponents and Surds:

[tex]\boxed { \sqrt { x } = x ^ { \frac{1}{2} } }[/tex]

[tex]\boxed { (a ^ b) ^ c = a ^ { b . c } } [/tex]

[tex]\boxed {a ^ b \div a ^ c = a ^ { b - c } }[/tex]

[tex]\boxed {\log a + \log b = \log (a \times b) }[/tex]

[tex]\boxed {\log a - \log b = \log (a \div b) }[/tex]

Let us tackle the problem.

[tex]\sqrt{(x-1)^3} = 8[/tex]

[tex](x-1)^3 = 8^2[/tex]

[tex](x-1)^3 = (2^3)^2[/tex]

[tex](x-1)^3 = 2^6[/tex]

[tex]\sqrt[3]{(x-1)^3} = \sqrt[3]{2^6}[/tex]

[tex](x - 1 ) = 2^{6/3}[/tex]

[tex]x - 1 = 2^2[/tex]

[tex]x - 1 = 4[/tex]

[tex]x = 4 + 1[/tex]

[tex]\large {\boxed {x = 5} }[/tex]

[tex]\sqrt[4]{(x-3)^5} = 32[/tex]

[tex]\sqrt[4]{(x-3)^5} = 2^5[/tex]

[tex](x-3)^5 = (2^5)^4[/tex]

[tex](x-3) = \sqrt[5]{2^{20}}[/tex]

[tex](x-3) = 2^4[/tex]

[tex](x-3) = 16[/tex]

[tex]\large {\boxed {x = 19} }[/tex]

[tex]\sqrt{(x-4)^3} = 125[/tex]

[tex]\sqrt{(x-4)^3} = 5^3[/tex]

[tex](x-4)^3 = (5^3)^2[/tex]

[tex](x-4)^3 = 5^6[/tex]

[tex]\sqrt[3]{(x-4)^3} = \sqrt[3]{5^6}[/tex]

[tex](x - 4 ) = 5^{6/3}[/tex]

[tex]x - 4 = 5^2[/tex]

[tex]x - 4 = 25[/tex]

[tex]x = 25 + 4[/tex]

[tex]\large {\boxed {x = 29} }[/tex]

[tex]\sqrt[3]{(x+2)^4} = 16[/tex]

[tex]\sqrt[3]{(x+2)^4} = 2^4[/tex]

[tex](x+2)^4 = (2^4)^3[/tex]

[tex](x+2) = \sqrt[4]{2^{12}}[/tex]

[tex](x+2) = 2^3[/tex]

[tex](x+2) = 8[/tex]

[tex]\large {\boxed {x = 6} }[/tex]

Learn more

  • Coefficient of A Square Root : https://brainly.com/question/11337634
  • The Order of Operations : https://brainly.com/question/10821615
  • Write 100,000 Using Exponents : https://brainly.com/question/2032116

Answer details

Grade: High School

Subject: Mathematics

Chapter: Exponents and Surds

Keywords: Power , Multiplication , Division , Exponent , Surd , Negative , Postive , Value , Equivalent

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