Math Problem Statement

Solve the equations: 3^{-3k} * 3^{-k} = 3^{-k}, 6^{-2b} * 6^2 = 6^{-3b}, 3^{2n} = 3^{n+2}.

Solution

Let me transcribe and solve the equations from the image.

The problems given are:

65) 33k3k=3k3^{-3k} \cdot 3^{-k} = 3^{-k}

66) 62b62=63b6^{-2b} \cdot 6^2 = 6^{-3b}

67) 32n=3n+23^{2n} = 3^{n+2}

Solution for each equation:


65) 33k3k=3k3^{-3k} \cdot 3^{-k} = 3^{-k}

  • First, use the product rule for exponents: aman=am+na^m \cdot a^n = a^{m+n}.

33k3k=33kk=34k3^{-3k} \cdot 3^{-k} = 3^{-3k - k} = 3^{-4k}

Now the equation becomes:

34k=3k3^{-4k} = 3^{-k}

Since the bases are the same, equate the exponents:

4k=k-4k = -k

Divide both sides by k-k:

4=14 = 1

There seems to be no solution for this equation since it leads to a contradiction. Therefore, no solution.


66) 62b62=63b6^{-2b} \cdot 6^2 = 6^{-3b}

  • Again, use the product rule for exponents:

62b62=62b+26^{-2b} \cdot 6^2 = 6^{-2b + 2}

The equation now becomes:

62b+2=63b6^{-2b + 2} = 6^{-3b}

Since the bases are the same, equate the exponents:

2b+2=3b-2b + 2 = -3b

Solve for bb:

2=b2 = -b

b=2b = -2

So, the solution is b=2b = -2.


67) 32n=3n+23^{2n} = 3^{n+2}

  • Since the bases are the same, equate the exponents:

2n=n+22n = n + 2

Solve for nn:

2nn=22n - n = 2

n=2n = 2

So, the solution is n=2n = 2.


Summary of Solutions:

  • 65) No solution.
  • 66) b=2b = -2
  • 67) n=2n = 2

Let me know if you want further details on these solutions or if you have any questions!

Here are 5 related questions to extend your practice:

  1. Solve: 2x+122x=242^{x+1} \cdot 2^{2x} = 2^4.
  2. Solve: 53a5a=575^{3a} \cdot 5^{a} = 5^{7}.
  3. Solve: 102b10b=10310^{-2b} \cdot 10^{b} = 10^{-3}.
  4. Solve: 42n=4n+34^{2n} = 4^{n+3}.
  5. Solve: 7k172k=757^{k-1} \cdot 7^{2k} = 7^{5}.

Tip: When solving exponential equations, always check if the bases can be made the same—this is the key to simplifying and equating exponents.

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Math Problem Analysis

Mathematical Concepts

Exponential Equations
Properties of Exponents

Formulas

Product of Powers: a^m * a^n = a^(m+n)
Equating Exponents: If a^m = a^n, then m = n

Theorems

Exponentiation Rules

Suitable Grade Level

Grades 8-10